COMPETITIVE ADSORPTION OF FURFURAL AND PHENOLIC …
Transcript of COMPETITIVE ADSORPTION OF FURFURAL AND PHENOLIC …
Journal of Engineering Volume 13 September2006 Number 3
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COMPETITIVE ADSORPTION OF FURFURAL AND PHENOLIC
COMPOUNDS ONTO ACTIVATED CARBON IN FIXED BED
COLUMN
PROF. DR. ABBAS H. SULAYMON
DR.KAWTHER W. AHMED Environmental Engineering Department-college of Engineering-University of Baghdad, Iraq
ABSTRACT
For a multicomponent competitive adsorption of furfural and phenolic compounds, a
mathematical model was built to describe the mass transfer kinetics in a fixed bed column with
activated carbon. The effects of competitive adsorption equilibrium constant, axial dispersion, external
mass transfer and intraparticle diffusion resistance on the breakthrough curve were studied for weakly
adsorbed compound (furfural) and strongly compounds (parachlorophenol and phenol). Experiments
were carried out to remove the furfural and phenolic compound from aqueous solution. The
equilibrium data and intraparticle diffusion coefficients obtained from separate experiments in a batch
absorber, by fitting the experimental data with theoretical model. The results show that the
mathematical model includes external mass transfer and pore diffusion using nonlinear isotherms,
provides a good description of the adsorption process for furfural and phenolic compounds in fixed
bed adsorber.
KEY WORDS: Competitive, Adsorption, Fixed-Bed, Activated-Carbon, Multi-component,
Mathematical Model, Mass Transfer Coefficient.
INTRODUCTION
Furfural and phenolic compounds are organic compounds that enter the aquatic environment
through direct discharge from oil refineries. The content of these pollutants in the industrial
wastewater are usually higher than the standard limit (less than 5 ppm for furfural and less than 0.5
ppm for phenolic compounds).
Activated carbon adsorption is one of the important unit processes that is used in the treatment
of drinking waters and renovation of wastewaters (Alexander, 1989).
Understanding of the dynamics of fixed bed adsorption column for modeling is a demanding
task due to the strong nonlinearities in the equilibrium isotherms, interference effects of competition of
solute for adsorbent sites, mass transfer resistance between fluid phase and solid phase and fluid-
dynamics dispersion phenomena. The interplay of these effects produces steep concentration fronts,
which moves along the column during the adsorption process, which has to be accounted for in
modeling (Babu, 2004).
A. H. SULAYMON COMPETITIVE ADSORPTION OF FURFURAL AND
K.W. AHMED PHENOLIC COMPOUNDS ONTO ACTIVATED
Carbon in Fixed Bed Column
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Several rate models have been developed that take into account an external film transfer rate
step, unsteady state transport in the solid phase and nonlinear equilibrium isotherm to predict
adsorption rates in batch reactor and fixed bed (Crittenden and Weber, 1978).
The key parameters for design of the adsorption system are the process parameters that are used
for modeling the system for predicting the quality of effluent under a wide range of operating
conditions. The key process parameters in adsorption such as isotherm constants and mass transfer
coefficients are established by conducting batch studies of adsorption. Established isotherm models
such as Langmuir and Freundlich are used for assessing the suitability of an adsorbent in adsorption
system, where the experimental data are fitted to any one of these models.
The parameters that are responsible for mass transfer operation are the external mass transfer
coefficient and intraparticle diffusivity or surface diffusion coefficient in the case of the homogeneous
solid phase diffusion model.
The objective of the present research are to conduct experiments on the competitive adsorption
equilibrium and adsorption kinetics in a fixed bed for removal of furfural (Fu) in the presence of
phenolic compounds (Ph, PCP) from aqueous solution and to compare the experimental results with
that simulated by the numerical solution of the general rate model which include axial dispersion, film
mass transfer, pore diffusion resistance and nonlinear isotherms.
MODELING OF MULTICOMPONENT FIXED-BED ADSORBER
The dynamics of a fixed bed is described by a set of convection diffusion equations, coupled
with source terms due to adsorption and diffusion inside the adsorbent particles.
The adsorption column is subjected to axial dispersion, external film resistance and intraparticle
diffusion resistance.
A rate model which considers axial dispersion, external mass transfer, intraparticle diffusion
and nonlinear isotherms is called a general multicomponent rate model. Such a model is adequate in
many cases to describe the adsorption and mass transfer processes in multicomponent adsorption
(Eggers, 2000; Volker, 1999).
The following equations are based on the hypothesis of an intraparticular mass transfer
controlled by diffusion into macropores (pore diffusion model). This approach considers three phases:
a. The mobile phase flowing in the space between particles.
b. The stagnant film of mobile phase immobilized in the macropores.
c. The stationary phase where adsorption occurs.
The following basic assumptions are made in order to formulate a general rate model (Eggers,
2000):
Adsorption process is isothermal.
The adsorbent particles in the column are spherical and uniform in diameter.
The concentration gradients in the radial direction are negligible.
An instantaneous local equilibrium exists between the macropore surface and the
stagnant fluid inside macropores of the particles.
The film mass transfer mechanism can be used to describe the interfacial mass transfer
between the bulk-fluid and particle phases.
The diffusional and mass transfer parameters are constant and independent of the
mixing effects of the components involved.
Continuity equation in the bulk-fluid phase:
01
2
2
t
q
t
C
Z
C
Z
CD i
b
bbibibi
bi
…1
Using Cpi, the concentration in the stagnant fluid-phase (in the macropores), and writing the
expression of interfacial flux leads to:
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pRRpibi
p
fii CCR
k
t
q
,
3 …2
Substitution of equation 2 into equation 1 gives:
013
,2
2
pRRpibi
pb
bfbibibibi CC
R
k
t
C
Z
C
Z
CD
…3
The particle phase continuity equation in spherical coordinates is:
01
1 2
2
*
R
CR
RRD
t
C
t
C pi
pip
pi
p
pi
p …4
Initial and boundary conditions
0),0( ZCC bibi …5
0),,0( ZRCC pipi …6
Z = 0: oibi
bi
bi CCDZ
C
…7
Z =L: 0
Z
Cbi …8
R = 0: 0
R
C pi …9
R = Rp: pRRpibi
pip
fipiCC
D
k
R
C
,
…10
Defining the following dimensionless variables;
oi
bi
biC
Cc ,
oi
pi
piC
Cc ,
oi
pi
piC
Cc
*
* , L
t ,
pR
Rr ,
L
Zz
bi
LiD
LPe
,
pip
pfi
iD
RkBi
,
2
p
pip
iR
LD ,
b
bii
i
Bi
13
The model equations can be transformed into the following dimensionless equations:
01
1,2
2
rpibii
bibibi
Li
ccc
z
c
z
c
Pe
…11
01
1 2
2
*
r
cr
rrcc
pi
ipippip
…12
Initial condition becomes ( = 0):
A. H. SULAYMON COMPETITIVE ADSORPTION OF FURFURAL AND
K.W. AHMED PHENOLIC COMPOUNDS ONTO ACTIVATED
Carbon in Fixed Bed Column
1724
0),0( zcc bibi …13
0),,0( zrcc pipi …14
And boundary conditions become;
z = 0: 1
biLi
bi cPez
c …15
z =1: 0
z
cbi …16
r = 0: 0
r
c pi …17
r = 1: 1,
rpibii
piccBi
r
c …18
The concentration cpi* in equation 12 is the dimensionless concentration of component i in the
solid phase of the particles. It is directly linked to a multicomponent Langmuir isotherm:
ss N
j
pjj
pii
N
j
pjj
piii
piie
Cb
Ca
Cb
CbqCq
11
*
,
11
…19
And in dimensionless form:
sN
j
pjojj
pii
pi
cCb
cac
1
*
1
…20
Finite element method is used for the discretization of the bulk-fluid phase partial differential
equation and the orthogonal collocation method for the particle phase equations an ordinary
differential equation system is produced. The ordinary differential equation system with initial values
can be readily solved using an ordinary differential equation solver such as the subroutine "ODE15S"
of MATLAB which is a variable order solver based on the numerical differentiation formulas (NDFs).
Optionally it uses the backward differentiation formulas (BDFs), which is also known as Gear's
method.
EXPERIMENTAL WORK AND PROCEDURE
The granulated activated carbon (GAC) used in the experiments was supplied by Unicarbon,
Italian. Its physical properties are listed in table 1.
The GAC was sieved into 28/32 mesh with geometric mean diameter of 0.5 mm. The GAC was
boiled, washed three times in distilled water and dried at 110o C for 24 hours, before being used as
adsorbent.
The aqueous solutions of furfural, phenol and parachlorophenol where prepared using reagent
grades. Their properties are listed in table 2.
The experiments were adjusted at the initial pH of 5.7 for phenolic compounds (Ping and
Guohua (I), 2001) and 8.1 for furfural, with 0.01 mol/l NaOH and 0.01 mol/l HCl. This is for single
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component experiments, while for multicomponent systems, the pH were adjusted at 7. Solutions were
not buffered to avoid adsorption competition between organic and buffer (Monneyron and Faur-
Brasqet, 2002).
The experiments were carried out in Q.V.F. glass column of 50 mm (I.D.) and 50 cm in height.
The GAC was confined in the column by fine stainless steel screen, at the bottom and glass packing at
the top of the bed to ensure a uniform distribution of influent through the carbon bed. The influent
solutions were introduced to the column through a perforated plate, fixed at the top of the column.
Feed solutions were prepared in Q.V.F. vessel supplied with immersed heater with a thermocouple to
adjust the temperature of the solution.
For the determination of adsorption isotherms, 250 ml flasks were filled with 100 ml of known
concentration of solutes and a known weight of GAC. The flasks were then placed on a shaker and
agitated continuously for 30 hours at 30o C. The concentration of furfural, phenol and
parachlorophenol in the solutions were determined by a UV-160A spectrophotometer at 254, 270 and
300 nm, respectively.
The adsorbed amount is calculated by the following equations:
A
eoe
W
CCVq
…21
The intraparticle diffusion coefficient for each solute was obtained by 2 liter Pyrex beaker fitted
with a variable speed mixer. The beaker was filled with 1 liter of known concentration solution and the
agitation started before adding the GAC. At time zero, the accurate weight of GAC where added.
Samples were taken every 5 minutes.
The necessary dosage of GAC, to reach equilibrium related concentration of Ce/Co equal 0.05,
were calculated from isotherms model and mass balance equation as follow:
e
eoA
q
CCVW
…22
Where:
ee Cfq …23
RESULTS AND DISCUSSION
Adsorption isotherm
The equilibrium isotherms for the investigated solutes onto GAC are presented in figure 1 .All
the adsorption isotherm display a nonlinear dependence on the equilibrium concentration.
The adsorption data for all the system fitted by Langmuir (Lucas and Cocero, 2004), Freundlich
(Weber and Walter, 1972), Radke-Prausnitz (Radke and Prausnitz, 1972), Dubinin-Radushkevich
(Monneyron and Faur-Brasqet, 2002), Reddlich-Peterson (Jossens et al., 1972) and combination of
Langmuir-Freundlich models (Sips, 1984).
The correlation between experimental data and the theoretical models was very good for all
systems. The Langmuir adsorption model was selected to be introduced in the fixed bed model, where:
e
eme
bC
bCqq
1 …24
A. H. SULAYMON COMPETITIVE ADSORPTION OF FURFURAL AND
K.W. AHMED PHENOLIC COMPOUNDS ONTO ACTIVATED
Carbon in Fixed Bed Column
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The equilibrium data for furfural, phenol and parachlorophenol (multicomponent system)
aqueous solution in GAC is adapted as the competitive Langmuir isotherm model (Fahmi and
Munther, 2003):
sN
j
jej
ieiim
ie
Cb
Cbqq
1
,
,,
,
1
…19
In which the parameters qm,i, bi are the coefficient of the single component. The parameters are
evaluated to be:
qm, Fu= 0.3744 kg/kg bFu=18.42 m
3/kg
qm, Ph= 0.3500 kg/kg bPh=34.6 m3/kg
qm, PCP= 0.3199 kg/kg bPCP=49.6 m3/kg
INTRAPARTICLE DIFFUSION COEFFICIENT
There were a good matching between batch experimental results and predicted data using pore
diffusion model for batch operation (Ping and Guohua (II), 2001) as shown in figure 2.
The pore diffusion coefficient for each solute are evaluated form the batch experiments to be
Dp, Fu = 9.870×10-10
m2/s, Dp, Ph = 8.251×10
-10 m
2/s and Dp, PCP =7.657×10
-10 m
2/s.
The external mass transfer coefficients in packed bed model for each solute were evaluated by
the correlation of Wilson and Gearkoplis (Ping and Guohua (I), 2001).
31
31
Re09.1
i
b
i ScSh
for Re=0.0015-55 …25
Where: im
pif
iD
dKSh
,
, ,
imw
wi
DSc
,
and
w
pwud
Re .
In which the molecular diffusion coefficient Dm,i of furfural, phenol and parachlorophenol in
aqueous solution are listed in table 2.
These values substitutes in equation 25 to evaluate Kf,i at different interstitial velocity in the
mathematical mode.
The axial dispersion coefficient calculated form Chung and Wen equation (Gupta et al., 2001):
48.0Re011.02.0
Re
w
wbD
…26
BREAKTHROUGH CURVE
Figures 3 to 10 show that the experimental and predicted breakthrough curves for
multicomponent system at different flow rate, bed depth and initial concentration of solutes at constant
temperature of 30o C. It is clear from these figures that:
1. The adsorption capacity order for the ternary system onto GAC as follow:
PCP > Ph > Fu
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In this case, the capacity of the adsorbate seems to influence the adsorption energy greatly. The
octanol-water partition coefficient Kow given in table 2, shows that the parachlorophenol is more
hydrophobic than phenol and furfural.
It has been clearly demonstrated that adsorption of phenolic compounds onto activated carbon
induces the formation of π-π bond, where activated carbon act as an electron donor and the solute
benzene ring has an electron-withdrawing character (Monneyron and Faur-Brasqet, 2002). The
mesomeric and/or inductive character of the substitutent of the aromatic compound influences this
formation and thus the molecule's adsorption energy. This interpretation is supported by experimental
results: the withdrawing inductive character of chloride substitutents (Gupta, 1946) decreases the
electron density of the parachlorophenol-benzene ring compared with that of phenol-benzene ring
(Wheeler and Levy, 1959). The adsorption energy of parachlorophenol is higher than that of Ph. These
results agree with Ping and Guohua (I) (2001).
The adverse effect of the OH group (Bartell and Miller, 1923) on adsorption of phenol may be
attributed to the capability of this group to form hydrogen bonding with the water which renders the
compound less liable to be adsorbed in compared with parachlorophenol.
The lower adsorption capacity of furfural may be explained by its higher solubility, low
molecular weight and low octanol-water coefficient in compared with parachlorophenol or phenol.
Furfural is a polar solvent and activated carbon is generally regarded to favor the adsorption of
non-polar compound rather than polar compounds. Since pure carbon surface is considered to be non-
polar, but in actual practice, some carbon-oxygen complexes are present which render the surface
slightly polar (Al-Bahrani and Martin, 1977).
2. In case of multicomponent systems, at the initial stage, there are a lot of active sites of GAC, the
strongly and weakly adsorbed components take the active sites freely. With increasing time, the
weakly adsorbed component is not easily adsorbed but moves a head with the bulk fluid and takes the
active sites first in the front part of the fixed bed. Because the strongly adsorbed component tends to
take the active sites instead of the weakly adsorbed component, it will displace the sites that had been
taken by the weakly adsorbed components. The result is that the local concentration of the weakly
adsorbed component within the fixed bed adsorber is higher (Ping and Guohua (I), 2001).
3. The frontal concentration profile of the breakthrough curves, in a fixed bed adsorber, is related to
the initial solutes concentration, Biot number (Bi) and Peclet number (Pe).
4. An increase in the flow rate at constant bed depth will increase the Biot number with slight increase
in Peclet number, Biot is the ratio of external mass transfer rate to the intraparticle mass transfer rate.
When Biot number is large (that is, the intraparticle mass transfer is the controlled step (Ping
and Guohua (I), 2001)), the break point will appear early, this is due to the decrease in contact time
between the solute and the adsorbent at higher flow rate.
5. An increase in bed depth at constant flow rate will increase the Peclet number at constant Biot
number, where Peclet number is the ratio of axial convection rate to the axial dispersion rate.
When Peclet number is small (that is, the effect of axial dispersion is not negligible (Ping and
Guohua (I), 2001)), the breakthrough curve become flat and the break point appear early.
6. The simulated breakthrough curves for adsorption of ternary system (furfural, phenol, and
parachlorophenol) onto activated carbon are in a close agreement with the experimental results. Thus,
the mathematical model which includes axial dispersion, film mass transfer, pore diffusion resistance
and nonlinear isotherms provides a good description of the competitive adsorption process in fixed bed
adsorber.
A. H. SULAYMON COMPETITIVE ADSORPTION OF FURFURAL AND
K.W. AHMED PHENOLIC COMPOUNDS ONTO ACTIVATED
Carbon in Fixed Bed Column
1728
CONCLUSIONS
This work has studied the characteristics of the competitive adsorption of furfural in the
presence of phenolic compounds in aqueous solution in a fixed bed adsorber. A mathematical model
which includes external mass transfer and pore diffusion using non-linear isotherm was investigated.
The results show that the solubility and hydrophobicity has a large influence on adsorption
capacity and energy respectively. These influences were confirmed by ternary adsorption.
The general rate model provides a good description of the adsorption process of furfural and
phenolic compounds in fixed bed adsorber onto activated carbon.
REFERENCES
Alexander, P. M. and Zayas, I.,(1989), "Particle size and shape effects on adsorption rate
parameters", J. Environmental Engineering, 115 (1), Feb., p 41-55.
Al-Bahrani, K. S. and Martin, R. J., (1977), Water Res., 11, pp 991-999.
Babu, B. V. and Gupta, S., (2004), "Modeling and simulation for dynamics of packed bed adsorption",
Chem. Conf., Mumbai.
Bartell, F. E. and Miller, E. J., (1923), J. Amer. Chem. Soc., 45, pp 1106-1110.
Crittenden, J. C. and Weber, I. R., (1978), "Predictive model for design of fixed-bed adsorbers, single
component model verification", Environmental Engineering Division, 104 (EE3), June, pp 433-
443.
Eggers, R., (2000), "Simulation of frontal adsorption", HIWI report by Hamburg-Hamburg University.
Fahmi, A. and Munther, K., (2003), "Competitive adsorption of Nickel and cadmium on sheep monure
waste, experimental and prediction studies", Separation Science and Technology, 38 (2), pp 483-
497.
Gu, T. and Zheng, Y., (1999), "A study of the scale-up of reversal-phase liquid chromatography",
Separation and Purification technology, 15, pp 41-58.
Gupta, A., Nanoti, O. and Goswami, A. N., (2001), "The removal of furfural from water by adsorption
with polymeric resin", Separation Science and Technology, 36 (13), pp 2835-2844.
Gupta, P., (1946), "Polarity of molecules in relation to their adsorption by charcoal", J. Indian Chem.
Soc., 23, pp 353-360.
Jossens, L., Prausnitz, J. M. and Frits, W., (1978), "Thermodynamic of multi-solute adsorption from
dilute aqueous solutions", Chem. Eng. Sci., 33, pp 1097-1106.
Lucas, S. and Cocero, M. J., (2004), "Adsorption isotherms for ethylacetate and furfural on activated
carbon from supercritical carbon dioxide", Fluid Phase Equilibria, 219, pp 171-179.
Monneyron, P. and Faur-Brasqet, C., (2002), "Competitive adsorption of organic micropollutant in the
aqueous phase onto activated carbon cloth", Langmuir, 18, pp 5163-5168.
Journal of Engineering Volume 13 September2006 Number 3
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Ping L. and Guohua, X. (I), (2001), "Competitive adsorption of phenolic compounds onto activated
carbon fibers in fixed bed", J. Environmental Engineering, August, pp 730-734.
Ping L. and Guohua, X. (II), (2001), "Adsorption and desorption of phenol on activated carbon fibers
in fixed bed", Separation Sience and Technology, 36 (10), pp 2147-2163., August, pp 730-734.
Radke, C. J. and Prausnitz, J. M., (1972), "Adsorption of organic compounds from dilute aqueous
solution on activated carbon", Ind. Eng. Chem. Fund., 11, pp 445-451.
Sips, R., (1948), J. Chem. Phys., 16, pp 490-495.
Volker, K., (1999), "Simulation of liquid chromatography and simulated moving bed system",
Technical report by Hamburg-Hamburg University (TUHH).
Weber, J. R. and Walter, J., (1972), "Physicochemical processes for water quality control", Wiley
Inter. Science, New York.
Wheeler, O. H. and Levy, E. M., (1959), Can. J. Chem., 37, pp 1235-1240.
NOTATION
Symbols
b Langmuir constant, m3/kg
Bi Biot number
pip
pf
D
RK
, -
C Concentration in fluid, kg/m3
Co Initial concentration, kg/m3
Db Axial dispersion coefficient, m2/s
Dm Molecular diffusion coefficient, m2/s
Dp Pore diffusion coefficient, m2/s
dp Particle diameter, m
Kf Fluid to particle mass transfer coefficient, m/s
L Length of column, m
Pe Peclet number
bD
uL, -
Q Fluid flow rate, m3/s
qe Internal concentration of solute in particle, kg/kg
qm Adsorption equilibrium constant defined by Langmuir equation, kg./kg
Re Reynolds number
w
pwud
, -
Rp Radius of particle, m
Sc Schmidt number
imw
w
D ,
, -
Sh Sherwood number
im
pif
D
dK
,
,, -
t Time, s
A. H. SULAYMON COMPETITIVE ADSORPTION OF FURFURAL AND
K.W. AHMED PHENOLIC COMPOUNDS ONTO ACTIVATED
Carbon in Fixed Bed Column
1730
Symbols
u Interstitial velocity
bpR
Q
2, m/s
V Volume of solution, m3
WA Mass of activated carbon, kg
Z Axial distance, m
Greek symbols
εb Bed porosity, -
εp Porosity of adsorbent, -
μw Viscosity of water, Pa.s
ρw Density of water, kg/m3
Subscript
b Bulk fluid phase
e Equilibrium
Fu Furfural
i Component number (1, 2, 3 …)
L Liquid phase
o Initial
p Particle phase
PCP Parachlorophenol
Ph Phenol
TABLES AND FIGURES
Table 1, Physical properties of activated carbon
Raw material Coconut shell
Apparent density, kg/m3 480 - 490
Bulk density, kg/m3 770
BET surface area, m2/g 1100
Particle porosity 0.5
Bed porosity 0.41
Iodine number, mg/g 1100 - 1130
pH 10.2 - 10.6
Ash content, % 5 (max)
Particle size, %
Mesh +12 0.2
12 – 16 20.67
16 – 20 25.05
20 – 30 34.18
30 – 40 18.83
40 - 1.08
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Table 2, Main properties of adsorbates
Property Furfural Phenol Parachlorophe
nol
Symbol Fu Ph PCP
Formula C5H4O2 C6H5OH C6H5OCl
Structure
O
O
OH
Cl
OH
Molecular weight, g/mole 96.08 94.11 129
Solubility in water (at 20o C), g/l 83 66.7 28
Log (Kow) 1.58 1.50 2.35
Molecular diffusion (at 20o C), m
2/s 1.04×10
-8 9.6×10
-9 8.82×10
-9
0.00
0.05
0.10
0.15
0.00 0.01 0.01 0.02 0.02 0.03 0.03
Ce (Kg/m3)
qe (
Kg
/Kg
)
PCP Isotherm
Phenol Isotherm
Furfural Isotherm
theoretical
Fig (1) Adsorption isotherm for PCP,Phenol and Furfural onto
activated carbon at 303 K.
A. H. SULAYMON COMPETITIVE ADSORPTION OF FURFURAL AND
K.W. AHMED PHENOLIC COMPOUNDS ONTO ACTIVATED
Carbon in Fixed Bed Column
1732
0.000
0.200
0.400
0.600
0.800
1.000
1.200
0 50 100 150 200 250
time (min.)
C/C
o
Furfural system
Phenol system
PCP system
Theoretical
Fig (2) Comparison of the measured concentration-time data
with that predicted by pore diffusion model in batch
adsorber for Furfural ,Phenol and PCP systems.
Fu-Ph-PCP adsorption Q=1.39*10-6m 3/s,L=0.25m
Cofu=0.05Kg/m 3,Co,ph=0.01Kg/m 3,Co,pcp=0.01Kg/m 3
0.0
0.3
0.6
0.9
1.2
1.5
0 50000 100000 150000 200000
Time (s)
C/C
o
Theoretical
PCP, Pe=105,Bipcp=141
Ph, Pe=105,Biph=138
Fu, Pe=105,Bifh=122
Fig (3) The experimental and predicted breakthrough curves
for adsorption of Fu-Ph-PCP system onto activated
carbon at Co,pcp=0.01 kg/m3.
Theoretical
PCP, Pe=105,Bipcp=141
Ph , Pe=105, Biph=138
Fu , Pe=105, Bifu=122
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Fu-Ph-PCP adsorption Q=1.39*10-6m 3/s,L=0.25m
Co,fu=0.05Kg/m 3,Co,ph=0.01Kg/m 3,Co,pcp=0.03Kg/m 3
0.0
0.3
0.6
0.9
1.2
1.5
0 20000 40000 60000 80000 100000 120000
Time (s)
C/C
o
Theoretical
PCP, Pe=105,Bipcp=141
Ph, Pe=105,Biph=138
Fu, Pe=105,Bifu=122
Fig (4) The experimental and predicted breakthrough curves
for adsorption of Fu-Ph-PCP system onto activated
carbon at Co,pcp=0.03 kg/m3.
Fu-Ph-PCP adsorption ,Q=1.39*10-6
m3/s,L=0.25m,
Co,fu=0.05Kg/m3,Co,pcp=0.02Kg/M
3,Co,ph=0.005Kg/m
3
0.0
0.3
0.6
0.9
1.2
1.5
0 20000 40000 60000 80000 100000 120000 140000 160000Time (s)
C/C
o
Theoretical
PCP, Pe=105,Bipcp=141
Ph, Pe=105,Biph=138
Fu, Pe=105,Bifu=122
Fig (5) The experimental and predicted breakthrough curves
for adsorption of Fu-Ph-PCP system onto activated
carbon at Co,ph=0.005 kg/m3.
Theoretical
PCP, Pe=105,Bipcp=141
Ph , Pe=105, Biph=138
Fu , Pe=105, Bifu=122
Theoretical
PCP, Pe=105,Bipcp=141
Ph , Pe=105, Biph=138
Fu , Pe=105, Bifu=122
A. H. SULAYMON COMPETITIVE ADSORPTION OF FURFURAL AND
K.W. AHMED PHENOLIC COMPOUNDS ONTO ACTIVATED
Carbon in Fixed Bed Column
1734
Fu-Ph-PCP adsorption ,Q=1.39*10-6
m3/s,L=0.25m,
Co,fu=0.05Kg/m3,Co,pcp=0.02Kg/m
3,Co,ph=0.02Kg/m
3
0.0
0.3
0.6
0.9
1.2
1.5
0 20000 40000 60000 80000 100000 120000 140000Time (s)
C/C
o
Theoretical
PCP , Pe=105,Bipc=141
Ph, Pe=105,Biph=138
Fu, Pe=105,Bifu=122
Fig (6) The experimental and predicted breakthrough curves
for adsorption of Fu-Ph-PCP system onto activated
carbon at Co,ph=0.02 kg/m3.
Fu-Ph-PCP adsorption,Q=1.39*10-6
m3/s,L=0.25m
Co,ph=0.01Kg/m3,Co,pcp=0.02Kg/m
3,Co,fu=0.01Kg/m
3
0.0
0.4
0.8
1.2
1.6
0 50000 100000 150000 200000
Time (s)
C/C
o
Theoretical
PCP, Pe=105,Bipcp=141
Ph, Pe=105,Biph=138
Fu, Pe=105,Bifu=122
Fig (7) The experimental and predicted breakthrough curves
for adsorption of Fu-Ph-PCP system onto activated
carbon at Co,fu=0.01 kg/m3.
Theoretical
PCP, Pe=105,Bipcp=141
Ph , Pe=105, Biph=138
Fu , Pe=105, Bifu=122
Theoretical
PCP, Pe=105,Bipcp=141
Ph , Pe=105, Biph=138
Fu , Pe=105, Bifu=122
Journal of Engineering Volume 13 September2006 Number 3
1735
Fu-Ph-PCP adsorption ,Q=1.39*10-6
m3/s,L=0.25m
Co,ph=0.01Kg/m3,Co,pcp=0.02Kg/m
3,Co,fu=0.2Kg/m
3
0.0
0.3
0.6
0.9
1.2
1.5
0 20000 40000 60000 80000 100000
Time (s)
C/C
o
Theoretical
PCP, Pe=105,Bipcp=141
Ph, Pe=105,Biph=138
Fu, Pe=105,BiFu=122
Fig (8) The experimental and predicted breakthrough curves
for adsorption of Fu-Ph-PCP system onto activated
carbon at Co,fu=0.2 kg/m3.
Fu-Ph-PCP adsorption,Q=1.39*10-6m3/s
Co,fu=0.05Kg/m3,Co,ph=0.01Kg/m3,Co,pcp=0.02Kg/m3
0.0
0.3
0.6
0.9
1.2
1.5
0 20000 40000 60000 80000 100000 120000 140000
Time (s)
C/C
o
Theoretical
Pe=105 ,Bifu=122,Biph=138,Bipcp=141
Pe=21,Bifu=122,Biph=138,Bipcp=141
Fig (9) The experimental and predicted breakthrough curves
for adsorption of Fu-Ph-PCP system onto activated
carbon at different bed depth.
Theoretical
PCP, Pe=105,Bipcp=141
Ph , Pe=105, Biph=138
Fu , Pe=105, Bifu=122
Theoretical
L=0.25 m
L=0.05 m
A. H. SULAYMON COMPETITIVE ADSORPTION OF FURFURAL AND
K.W. AHMED PHENOLIC COMPOUNDS ONTO ACTIVATED
Carbon in Fixed Bed Column
1736
Fu-Ph-PCP adsorption, L=0.25 m
Co,fu=0.05Kg/m3,Co,ph=0.01Kg/m3,Co,pcp=0.02Kg/m3
0.0
0.3
0.6
0.9
1.2
1.5
0 20000 40000 60000 80000 100000 120000 140000Time (s)
C/C
o
T heo ret ical
P e=105,B ifu=122,
P e=109,B ifu=176,
Fig (10) The experimental and predicted breakthrough curves
for adsorption of Fu-Ph-PCP system onto activated
carbon at different flow rate.
Theoretical
Q=1.39*10-6
m3/s
Q=4.17*10-6
m3/s
Journal of Engineering Volume 13 September2006 Number 3
1623
ANALYSIS OF GALVANIC CORROSION UNDER MASS
TRANSFER CONTROLLED CONDITIONS
Qasim. J.M Slaiman Basim O. Hasan Baker M. Al-Zaidy
Chemical Eng. Dept.- Nahrain University-Baghdad -Iraq
ABSTRACT
Because of practical importance of protecting industrial equipments from galvanic corrosion,
the need arises to analyze the effects of variables, such as temperature, velocity, and area fraction of
metals on galvanic corrosion in systems under mass transfer control as in seawater (pH=7). For these
reasons the galvanic corrosion of Fe-Zn is analyzed to study the influence of Reynolds number,
temperature, and area fraction on the galvanic corrosion rates and galvanic corrosion potential under
mass transfer control.
It is found that galvanic corrosion rate of more active metal (Zn) is increased with Reynolds
number while the corrosion rate of more noble metal (Fe) is slightly increased with Re depending on
the galvanic potential that depends on the area fraction. Increasing Reynolds number shifts the
galvanic potential to more positive values. Also increasing temperature leads to shift the corrosion
potential to more negative values and to change the corrosion rate of more active metal (Zn)
depending on two parameters oxygen solubility and oxygen diffusivity. As area fraction of more
active metal (Zn) increased the galvanic potential is shifted to the negative anodic direction while the
corrosion rate for more noble metal is decreased.
KEY WORDS: Galvanic corrosion, mass transfer control, Fe-Zn couple, temperature, area fraction
االخلاصة
الحاجة لدراسة وتحليل تأثير بعض العوامل ية من التاكل الغلفاني ظهرتعبسبب الاهمية التطبيقية لحماية المعدات الصنا
مثل درجة الحرارة وسرعة السائل ومساحة الكاثود والانود على التاكل الغلفاني في الانظمة التي تكون تحت سيطرة انتقال
كما في ماء البحر. (pH=7)الكتلة
تمت دراسة التاكل الغلفاني لمعدني الحديد والخارصين لمعرفة تاثير عدد رينولد )او السرعة( ودرجة الحرارة ونسبة
( Zn)المساحة في ظروف سيطرة انتقال الكتلة. أظهرت النتائج ان زيادة عدد رينولد يؤدي الى زيادة تأكل المعدن الفعال الانود
مساحة بين المعدنين. ل( وحسب جهد التاكل الذي يعتمد على نسبة اFeية ويؤثر قليلا على المعدن الاقل فعالية )الكاثودبصورة رئيس
زيادة درجة الحرارة تؤدي الى تقليل معدل التاكل الغلفاني و زيادة عدد رينولد يؤدي الى زيادة جهد التاكل الغلفاني بالاتجاه الموجب
تؤدي الى تقليل (Zn)زيادة مساحة المعدن الفعال اما وازاحته بالاتجاه السالب )ازاحته بالاتجاه السالب(فانيوتقليل جهد التاكل الغل
.(Fe) الجهد الغلفاني وتقليل تاكل المعدن الاقل فعالية
INTRODUCTION
Corrosion is the deterioration of materials by chemical interaction with their environment. The
term corrosion is some times also applied to the degradation of plastics, concrete, and wood but
generally refers to metals. The consequences of corrosion are many and varied and the effect of these
on the safe, reliable, and efficient operation of equipment or structures are often more serious than the
simple loss of a mass of metal. Failures of various kinds and the need for expensive replacements may
occur even though the amount of metal destroyed is quite small. Some of the major harmful effects of
Q. J.M Slaiman Analysis of Galvanic Corrosion Under
B. O. Hasan Mass Transfer Controlled Conditions
B.M. Al-Zaidy
4261
corrosion are [Shreir 2000]: reduction of metal thickness, hazards or injuries to people arising from
structural failure, loss of time, reduced value of goods, contamination of fluids in vessels and pipes,
perforation of vessels and pipes, loss of technically important surface properties of a metallic
component, and mechanical damage to valves, pumps, etc. Galvanic corrosion, often misnamed
"electrolysis," is one common form of corrosion in marine environments. It occurs when two (or
more) dissimilar metals are brought into electrical contact under corrosive environment. When a
galvanic couple forms, one of the metals in the couple becomes the anode and corrodes faster than it
would all by itself, while the other becomes the cathode and corrodes slower than it would alone.
Either (or both) metal in the couple may or may not corrode by itself (themselves) in seawater. When
contact with a dissimilar metal is made, however, the self-corrosion rates will cause the corrosion of
the anode to accelerate and corrosion of the cathode to decelerate or even stop. If any two metals are
coupled together, the one closer to the anodic (or active) end of the series, will be the anode and thus
will corrode faster, while the one toward the cathodic (or noble) end will corrode slower. The two
major factors affecting the severity of galvanic corrosion are, (i) the voltage difference between the
two metals on the galvanic series, (ii) the size of the exposed area of cathodic metal relative to that of
the anodic metal.
Corrosion of the anodic metal is more rapid and more damaging as the voltage difference
increases. It is well known that the rate-controlling step in most aerated water corrosion processes is
the cathodic half reaction. The most important cathodic process in aerated waters is oxygen reduction.
The rate of this half reaction is generally limited by the speed at which oxygen can reach the surface
of the metal. This oxygen is transported from the bulk water to the surface across the boundary layer
by diffusion [Smith et. al. 1989, Cheng and Steward 2004].
Many investigations were carried out to study the galvanic corrosion. Copson [1945] studied the
galvanic action between steel coupled to nickel in tap water with 3 to 1 area ratio of Ni/ Fe and found
that the galvanic corrosion of steel was appreciable. Pryor [1946] investigated the galvanic corrosion
of Al/steel couple in chloride containing solution and found that aluminum completely protects steel
cathodically within the pH range 0-14, and the galvanic current and the corrosion rate of aluminum
are at a minimum in the nearly neutral pH range. Wranglen et al. [1969] studied the difference
between the galvanic corrosion rates of high and low carbon steel in acid solutions and concluded that
the engineers should not depend only on the galvanic series in the selection of their materials of
construction. Mansffeld et al [1971, 1973a, 1973b, 1973c,1973d] investigated experimentally many
factors that affect the galvanic interaction of various metals and alloys (as Al and Ti) in 3.5% NaCl
solution and in HCl and gave a detailed explanations. Tsujino et al.[1982] studied the galvanic
corrosion of steel coupled to noble metals (Pt, Cu, 304 stainless steel) in sodium chloride solution and
found that the local currents on the steel depend on the area ratio of the steel to the cathodic metal and
these currents are not related to the concentration of sodium chloride in neutral solutions. Bardal et
al.[1984] predicted galvanic corrosion rates by means of numerical calculation and experimental
models based on boundary element method. Glass and Ashworth [1985] perform experimental study
to determine the corrosion rates of zinc- mild steel couple at 65 oC in pH of 8. They determined and
discussed the variation of corrosion potential and corrosion rate with time. Fangteng et al.[1988]
presented a theoretical approach for galvanic corrosion allowing for cathode dissolution, and found
that the cathode of the couple is also corroded at the galvanic corrosion potential where the corrosion
is controlled by the rate of oxygen diffusion to the electrode surfaces and the cathode dissolution in a
galvanic system leads to a decrease in the galvanic current and it has been shown that the current
density through the anode is independent of the area ratio of the electrodes. Jones and Paul [1988]
stated that many semi conducting minerals have sufficient conductivity to permit electrochemical
reactions on their surfaces and consequently, galvanic interactions will occur when such minerals are
coupled to metals or other conducting minerals. Morris and Smyrl [1989] calculated galvanic currents
and potentials on heterogeneous electrode surfaces comprised of random configurations of coplanar
anodes and cathodes, for the purpose of investigating system behavior on different electrode
geometries. Symniotis [1990] investigated the active dissolution of a duplex stainless steel in two
Journal of Engineering Volume 13 September2006 Number 3
1625
different acidic solutions together with a comparison of the active dissolution of the corresponding γ
and α phases and found that the galvanic action takes place between the two phses. Also he studied the
influence of dependence of anodic currents on time, surface morphology, and surface area. Chang et
al [1997] studied the galvanic corrosion behavior of tungsten coupled with several selected
metals/alloys. They stated that from an environmental perspective, tungsten is a more desirable
material than depleted uranium (DU) for penetration applications. Lee et al [2000] investigated the
corrosion behavior of an as-cast magnesium alloy focusing on the galvanic corrosion between a
precipitate and Mg-rich matrix. Al-Hadithi [2002] investigated experimentally the effects of
temperature, pH, and area fraction on the galvanic corrosion rate of binary galvanic system by
coupling carbon steel, zinc, copper, and brass under activation control conditions. He studied the
coupling of each pair of these metals individually. Song et al [2004] investigated experimentally the
galvanic corrosion of megnisium alloy AZ9ID in contact with zinc, aluminum, and steel alloys. AL-
Maypof [2006] studied the galvanic coupling between magnetite and iron in acidic solution.
The present study aims to analyze the galvanic corrosion behavior of binary metals (Fe-Zn)
under mass transfer control to investigate the influence of temperature, Reynolds number, and area
fraction, on the free corrosion rate and galvanic corrosion rate for binary galvanic system under mass
transfer (diffusion) control.
ANALYSIS
When two different metals are in a corrosive environment, they corrode at different rates
according to their specific corrosion resistances to that environment, however, if the two metals are in
contact, the more corrosion prone (metal 1) corrodes faster and the less corrosion prone (metal 2 the
more noble one) corrodes slower than originally, i.e. when no contact existed. The accelerated damage
to the less resistant metal is called galvanic corrosion, and is heavily dependent on the relative surface
areas of the metals
To determine the potential of a system in which the reduced and oxidized species are not at
unit activity, the familiar Nernest equation can be employed:
oxd
red
oa
aln
nF
RTEE (1)
Tafel slopes (Tafel constants) are determined from the following equation [Shreir 2000]
nFα
RTβ
a
a (2)
nF
RTβc
cα (3)
The relationship between reaction rate and overvoltage for activation polarization is
o
A
i
iβlogη (4)
The reaction rate is given by the reaction current or current density [David and James 1998]: )/E(Ea
ao,aaaeeii
(5)
and )/E(E
co,cceceii c
(6)
The effect of temperature is to change the value of exchange current density io as follows [Nesic et al
1996]:
Q. J.M Slaiman Analysis of Galvanic Corrosion Under
B. O. Hasan Mass Transfer Controlled Conditions
B.M. Al-Zaidy
4262
)]T
1
298
1(
R
Eexp[ii act
o,298To, (7)
The anodic current is given by [West 1965]
)](ERT
exp[Ai ,aa
aao, aea
a EFn
I
(8)
or
)](ERT
exp[fi ,a
a
aao, ae
a
a EFn
i
(9)
and cathodic one
)](ERT
exp[-Ai ,c
c
cco, ce
c
c EFn
I
(10)
)](ERT
exp[-fi ,c
c
cco, ce
c
c EFn
i
(11)
For diffusion (mass transfer) controlled corrosion systems the reaction current is given by Fick's law
[Shreir 2000]
)Ck(Cdx
dCD
FAz
Isb
c
(12)
The limiting current, i.e., the maximum current under diffusion control is obtained when Cs=0, so
bcL FAkCzI (13)
Where the mass transfer coefficient, k, is defined by, k=D/δ. The corrosion current is then
bcLcorr FAkCzII
or bcL FkCzi (14)
zc is used because in the corrosion processes the cathodic reaction is the one likely to be controlled by
diffusion. The bulk concentration of oxygen in the solution changes with temperature as shown in
Table 1.
The mass transfer coefficient (k) varies with flow or relative speed between metal and
environment, the geometry of system, and the physical properties of the liquid. To calculate k in
dynamic environment, the dimensionless groups are often used. Over the years there were many
correlations proposed for predicting k for systems under mass transfer control. The well known
correlation is that of Poulson and Robinson [1986] under turbulent flow conditions:
Sh=0.026Re0.82
Sc0.35
(15)
Hence the expression of k is
k=(D/d) 0.026Re0.82
Sc0.35
(16)
Journal of Engineering Volume 13 September2006 Number 3
1627
The effect of temperature and pressure on the diffusion coefficient is given by [Brodkey and Hershey
1989]
n
o
ooTP
T
T
P
PDD )(, (17)
where the exponent n varies from 1.75 to 2.
For galvanic corrosion under mass transfer or activation control at galvanic potential (Eg):
cIaIcorrI (18)
and
ca II (19)
For two metal galvanic corrosion
ia,1A1+ia,2A2= ic,1A1+ic,2A2 (20)
or ia,1f1+ia,2f2= ic,1f1+ic,2f2 (21)
Where f1 and f2 are the area fractions (individual metal area/total metals area) of metals 1 and 2
respectively. At galvanic corrosion potential, Ea= Ec= Eg, hence at Eg
)](ERT
exp[i 1,g
a
1ao,1 ae
a
a EFn
i
(22)
)](ERT
exp[i 2,g
a
2ao,2 ae
a
a EFn
i
(23)
)](ERT
exp[-i 1,gc
c1o,1 cec
c EFn
i
(24)
)](ERT
exp[-i 2,g
c
c2o,2 ce
c
c EFn
i
(25)
For mass transfer control the summation of cathodic currents equal the oxygen limiting diffusion
currents on both metals since the hydrogen evolution currents are negligible (pH=7). Hence
LΣIaΣI (26)
ia,1f1+ia,2f2= iLf1+iLf2 (27)
since f1+f2=1, hence
ia,1f1+ia,2f2= iL (28)
Insertion of Eqs. (22) and (23) in Eq.(28) with iL from Eqs. (14) and (16), Eg can be obtained by
iteration method.
Simplifications leading to analytic solutions of the above equations are so complex, so
numerical solutions must be attempted. As an example, a numerical method implemented on a
microcomputer. The sweeping method is as follows:
a. Estimate equilibrium potentials for metals using equation (1) at a particular temperature for pH of 7.
The calculations are performed for the activity of oxidized species (F2+
and Zn2+
) of 10-6
molar.
Q. J.M Slaiman Analysis of Galvanic Corrosion Under
B. O. Hasan Mass Transfer Controlled Conditions
B.M. Al-Zaidy
4261
b. Tafel slopes for anodic and cathodic reactions are established from equations (2) and (3) with αa =
αc =0.5.
c. The exchange current density is calculated from equation (7) for three values of temperatures 25, 40
and 60 oC with Eact from Table 2. Table 3 gives values of io at 25
oC.
d. Bulk concentration of oxygen in water at any temperature is from Table (1).
e. The value of oxygen diffusivity is estimated from Eq. (17) at different temperatures 25, 40 and 60 oC with oxygen diffusivity at 25
oC (D
o) of 2.04 × 10
-9 m
2/s [Perry and Green 1997].
f. The mass transfer coefficient k is calculated by using Eq. (16) with d=5 cm.
g. The limiting current is estimated of from Eqs. (14) and (16) at a particular Re at each temperature.
To calculate Schmidt number (Sc=ν/D) the physical properties of water are taken from Perry and
Green [1997].
h. Eg is assumed to start the iteration. It is necessary to realize that the galvanic corrosion potentials
(Eg) of the reactions involved are chosen between the more negative equilibrium potential and the less
negative one.
i. The values of Ee, ia, and Eg are substituted in Eqs.(22) and (23) to determine anodic currents.
j. The summations of the anodic and limiting currents (cathodic currents) are compared to determine
the absolute value of their difference.
k. A new value of Eg is assumed as in h until the difference between the summation of the anodic and
cathodic (limiting) currents becomes very small to obtain the galvanic corrosion potential.
RESULTS AND DISCUSSION
Free Corrosion
Fig. 1 shows the variation of free corrosion potential with Re for the two metals Fe and Zn at
25 oC. The Fig. indicates that increasing Re shifts the corrosion potential to more positive direction for
both metals. Fig. 2 shows the variation of Fe free corrosion potential with Re at different
temperatures. The Fig. reveals that increasing Re shifts the free corrosion potential to more positive
values. This can be ascribed to the increased oxygen transport to the metal surface. Also the Fig.
indicates that increasing temperature shifts the free corrosion potential to more negative due to the
decreased oxygen solubility. Fig.3 shows the variation of oxygen limiting current density with Re at
various temperatures. The oxygen limiting diffusion current equals the total cathodic currents since
the hydrogen evolution current is negligible in systems of PH=7. Increasing Re increases the limiting
current density via increasing oxygen supply to the metal surface by eddy diffusion [Fontana and
Green 1984, Poulson and Robinson 1986]. Also the figure reveals that the higher the temperature is
the lower the iL because the O2 solubility in the bulk of the solution decreases with temperature.
Galvanic Corrosion
Fig. 4 shows the variation of galvanic corrosion potential with Re at various area fraction
values. The Fig. shows that increasing Re leads to increase the galvanic potential to more positive
direction for the whole range of area fractions. This is ascribed to the fact that increasing Re leads to
increase the supply of cathodic species (oxygen) and hence increase the oxygen limiting diffusion
current density shifting the corrosion potential to more positive. Also at a particular Re the higher the
area fraction of Fe is the more positive value of galvanic corrosion potential. This trend holds for the
whole range of temperature. Fig. 5 for area fraction of Fe of 0.1 and Zn of 0.9 shows the variation of
current density with Re. It is evident that the Zn corrosion current density is equal to the oxygen
limiting current density while the Fe corrosion current density is very low or negligible indicating that
at coupling Fe with Zn, the Fe is totally protected because the equilibrium potential of Fe is higher
than the galvanic potential of Fe= Zn couple. Also the galvanic corrosion current density of Zn
increases appreciably with Re while that of Fe is not affected with Re since it is galvanically protected
with Zn because coupling Zn with Fe shifts the galvanic potential below the equilibrium potential of
Fe stopping its corrosion . Mansfeld [1971] stated that the corrosion rate of more active metal in
Journal of Engineering Volume 13 September2006 Number 3
1629
aerated neutral solutions is controlled by the diffusion rate of the oxidizer (oxygen) to the metal
surface. Fig. 6 for area fraction of Fe and Zn of 0.5 exhibits the same trend. Fig. 7 shows that
decreasing area fraction of Fe (or increasing area fraction of Zn) leads to decrease the Fe galvanic
corrosion current density indicating that increasing the area fraction of more active metal leads to
decrease the corrosion rate of more noble metal. Also at high Fe area fraction, increasing Re increases
the corrosion current density of Fe. Fig. 8 indicates that the Zn corrosion current density increases
with Re and slightly affected with the area fraction.
Figs. 9 and 10 for Re=10000 and 60000 respectively show the variation of galvanic corrosion
potential with temperature at various area fractions. The figures reveal that increasing the temperature
shifts the corrosion potential to more negative values. This behavior is attributed to certain solubility
considerations. Many gases such as oxygen have lower solubility in open systems at higher
temperatures. As temperature increases, the resulting decrease in solubility of gas causes corrosion
potential and corrosion rate to go down [Nesic et al. 1996, Shreir 2000]. The effect of Re on the Fe
galvanic corrosion current density in galvanic coupling with Zn at various temperatures is shown in
Figs. 11 and 12 for different area fractions. The figures reveal also that the Fe galvanic current density
varies with temperature where the highest corrosion current occurs at 40 oC and the lowest at 60
oC
depending on two parameters, oxygen solubility and oxygen diffusivity.
Practically Zn in certain environmental conditions may exhibits a passivity or polarity reversal
when coupled with Fe as noticed by Glass and Ashworth [1985] in 0.01 M NaHCO3 at 65 oC.
CONCLUIONS
1- At coupling two metals in systems under mass transfer control, the galvanic corrosion rate of
more active metal increases with increasing Re. The galvanic corrosion rate of more noble
metal is slightly affected by Re. The effect of Re on both metal is depending on area fraction
of both metals.
2- Increasing temperature shifts the galvanic corrosion potential to more negative while the effect
of temperature on the corrosion rate is unstable.
3- Increasing area fraction of more active metal has negligible effect on the corrosion rate of this
metal and decreases the corrosion rate for more noble metal via shifting the corrosion potential
to more negative.
Table 1 Solubility of Oxygen in Sea Water at 1 atm [Perry and Green 1997]
Temperature, oC Solubility of oxygen mg/l
25 7.8
40 6.0
60 3.1
Table 2: Activation Energy of Metals [West 1965]
Metal Activation Energy (J/mol)
Fe 2625
Zn 13609
Table 3: Values of Exchange Current Density at 25 oC and pH=7 [West 1965]
Metal Eo, V io, A/cm2
Fe -0.44 10-6
Zn -0.76 10-9
Q. J.M Slaiman Analysis of Galvanic Corrosion Under
B. O. Hasan Mass Transfer Controlled Conditions
B.M. Al-Zaidy
4261
Fig.1: Variation of Free Corrosion Potential with Re.
0 15000 30000 45000 60000 75000
Re
-0.2
-0.2
-0.1
-0.1
0.0
E ,
V (
SH
E)
Free Corrosion of Fe
25 C
40 C
60 C
co
rr
Fig.2: Variation of Free Corrosion Potential of Fe with Re
0 15000 30000 45000 60000 75000
Re
-0.9
-0.6
-0.3
0.0
0.3
E ,V
(S
HE
)
Free Corrosion , T=25 C
Fe
Zn
co
rr
Journal of Engineering Volume 13 September2006 Number 3
1631
0 20000 40000 60000
Re
0.00
2.00
4.00
6.00
8.00
i (
A/m
2)
T=25 C
T=40 C
T=60 C
L
Fig.3: Variation of Limiting Current Density with Re.
0 15000 30000 45000 60000
Re
-0.70
-0.65
-0.60
-0.55
-0.50
-0.45
E
Galvanic Corrosion, T=25 C
Area Fraction of Fe=0.1
Area Fraction of Fe 0.5
Area Fraction of Fe=0.9
Fig. 4: Variation of Galvanic Potential with Re for Fe-Zn Couple at T=25 C.
Q. J.M Slaiman Analysis of Galvanic Corrosion Under
B. O. Hasan Mass Transfer Controlled Conditions
B.M. Al-Zaidy
4266
0 15000 30000 45000 60000
Re
0.00
2.00
4.00
6.00
8.00
Curr
en
t D
ensity
(A/m
2]
Area Fraction Fe 0.1, T=25
Limiting Current
Fe galvanic Current
Zn galvanic Current
Fig. 5: Variation of Current Density with Re for Fe-Zn Couple at T=25 C.
0 15000 30000 45000 60000
Re
0.00
2.00
4.00
6.00
8.00
Curr
en
t D
ensity
(A/m
2]
Area Fraction Fe 0.5, T=25
Limiting Current
Fe galvanic Current
Zn galvanic Current
Fig. 6: Variation of Current Density with Re for Fe-Zn Couple at T=25 C.
0 15000 30000 45000 60000
Re
0
50
100
150
200
250
300
Fe
Galv
anic
Curr
en
t (
A/m
2]
T=25 C
Area Fraction of Fe=0.9
Area Fraction of Fe=0.5
Area fraction of Fe =0.1
10
×-9
Fig. 7: Variation of Current Density of Fe with Re at Various Area Fractions at T=25 oC.
Journal of Engineering Volume 13 September2006 Number 3
1633
0 15000 30000 45000 60000
Re
3
8
0
5
Zn G
alv
anic
Curr
en
t (
A/m
2]
T=25 C
Area Fraction of Fe=0.9
Area Fraction of Fe=0.5
Area fraction of Fe =0.1
Fig. 8: Variation of Current Density of Fe with Re for Various Area Fractions at T=25 oC.
20 30 40 50 60
T, C
-680
-640
-600
-560
-520
Eg
(m
v)
Re=10000
Area Fraction of Fe=0.1
Area Fraction of Fe=0.5
Area Fraction of Fe=0.9
Fig. 9: Variation of Galvanic Corrosion Potential with Temperature at Re=10000.
Q. J.M Slaiman Analysis of Galvanic Corrosion Under
B. O. Hasan Mass Transfer Controlled Conditions
B.M. Al-Zaidy
4261
20 30 40 50 60
T, C
-640
-600
-560
-520
-480
Eg
(m
v)
Re=60000
Area Fraction of Fe=0.1
Area Fraction of Fe=0.5
Area Fraction of Fe=0.9
Fig.10: Variation of Galvanic Corrosion Potential with Temperature at Re=6000.
0 20000 40000 60000
Re
0.00
1.00
2.00
3.00
4.00
5.00
Fe C
urr
en
t (A
/m2
) (
10
)
Area Fraction of Fe =0.1
T=25 C
T=40 C
T=60 C-9
Fig. 11: Variation of Fe Corrosion Current with Re at Various Temperatures for Fe Area
Fraction of 0.1.
0 20000 40000 60000
Re
0
1
2
3
4
Fe C
urr
ent
(A
/m2)
(1
0
)
Area Fraction of Fe =0.9
T=25 C
T=40 C
T=60 C-7
Fig. 12: Variation of Fe Corrosion Current with Re at Various Temperatures for Fe Area
Fraction of 0.9.
Journal of Engineering Volume 13 September2006 Number 3
1635
NOMENCLATURE
A Surface area of specimen m2
a Activities (concentration) of reduced and oxidized species Mol/liter
C Bulk concentration mole/m3
D Diffusion coefficient of reacting ion m2/s
d Diameter m
E Electrode potential V
F Faradays constant 96487 Coulomb/g.equivalent
i Current density µA/cm2
I Total corrosion µA
io Exchange current density at concentration µA/cm2
k Mass Transfer Coefficient m/s
L Distance between the pressure taps m
n Number of electrons transfer
R Gas constant 8.314 J/mol.K
Re Reynolds number
Sc Schmidt number.
T Temperature oC or K
u Velocity m/s
z Number of electrons Transferred
Greek Letters
α Symmetry factor
β Tafel slope V
δ Thickness of diffusion layer m
η overpotential V
μ Viscosity Kg/m.sec2
ν Kinematic viscosity m2/s
ρ Density Kg/m3
Subscripts
a anode
b bulk
c cathode
e equilibrium
g galvanic
L limiting
s surface
Abbreviations
corr corrosion
oxid oxidation
red reduction
Q. J.M Slaiman Analysis of Galvanic Corrosion Under
B. O. Hasan Mass Transfer Controlled Conditions
B.M. Al-Zaidy
4262
REFERENCE
Al- Hadithi F. F. (2002), " Computer Aided Simulation and Laboratory Investigation of Glavanic
Corrosion", Ph.D Theisis, Nahrain University, Baghdad, Iraq.
AL-Mayouf A.M., (2006)," Dissolution of megnettite coupled galvanically with iron in
environmentally friendly chelant solution, Corrosion Scince, 48, 898-912.
Bardal E., Johnson R., and Gastland P., Corrosion J., 12 40 (1984), P.628-P.633
Brodkey R. S. and H. C. Hershey, Transport Pkenomena, 2nd
Printing Mc Graw Hill, New York,
1989.
Chang, Beatty, Kane and Beck,, Tri-Service Conference on Corrosion. I; Naval Surface Warfare
Center-Carderock Division 1997, pp. 6.33-6.45.
Copson H.R., (1945) Ind. Eng. Chem. J., 8, 37, P.721-723.
David Tabolt and James Tabolt, 1998, "Corrosion Science and Technology", CRC series, Library of
Congress, 1st Edition, New York.
Fangteng, S., (1988) Corrosion Science Jounal, 6, 25, ,P.649-P.655.
M. G. Fontana, N. D. Green, 1984, Corrosion Engineering, 2nd
Edition.,London.
Glass G. K., and Y. Ashworth (19985), “The Behavior of the Zinc-Mild steel Galvanic Cell In Hot
Sodium Bicarbonate Solution”, Corrsion Science, Vol. 25, No. 11, pp. 971-983.
Jones D. A. and A. J. P. Paul, (1988) Corrosion 88/245. NACE, Houston, TX.
Lee , Kang , Shin, (2000) Metals and Materiald, Vol. 6, No. 4, pp. 351-358, Aug.
Mansfeld F., 1971, Corrosion Journal, 10, 27, pp. 436-442.
Mansfeld F., 1973a, Corrosion Journal, 7, 29, pp. 276-281.
Mansfeld F., 1973b, Corrosion Journal, 2, 29, pp. 56-58.
Mansfeld F., and Parry E. P., 1973c, Corrosion Journal, 4,13, pp. 397-402.
Mansfeld F., and Parry E. P., 1973d, Corrosion Journal, 10,30, pp. 343-353.
Morris, W. Smyrl, (1989) J. Electrochem. Soc., Vol. 136, No. 11, November, p. 3237-3248.
Nesic S., J. Postlethwaite , and S. Olsen, (1996) "An Electrochemical Model for Prediction of
Corrosion of Mild Steel in Aqueous Carbon Dioxide Solution", Corrosion J., April, Vol.52,
No.4, P.280.
Perry, R. H ,and Green ,D. W , 1997, Perry Chemical Engineers Handbook, 7th
ed, Mc Graw Hill
,United states ,
Poulson B. and R. Robinson (1986), "The Use of Corrosion Process to Obtain Mass Transfer
Correlations", Corr. Sci. Vol, 26, No.4, P.265.
Pryor, M.J., (1946) Corrosion Journal , Vol.1, P.14.
Shreir L.L, (2000), Corrosion, Vol. 1, Metal Environment Reactions, 3rd
Edition, Butterworth-
Heinemann.
Smith S. W., K. McCabe, and D.W. Black, 1989 Corrosion-NACE, 45, 790-793.
Song G., B. Johannesson, S. Hapugoda, and D. Stjohn, 2004," Galvanic Corrosion of Magnesium
Alloy in Contact with aluminum, Steel, and Zinc, Corr. Sci., 46, 995-977.
Symniotis E., (1990), "Galvanic Effects on the Active Dissolution of Duplex Stainless Steels",
Corrosion J., January, No.1, 46, P.2.
Tsujino, B .and Miyase S., (1982), Corrosion Journal, 4, 38, P.226-230.
West J. M. , (1965.), Electrodeposition and Corrosion Processes, Van Nostrand Co. LTD, London.
Wranglen G. and Khokhar , (1969), Corrosion Science Journal , 8, 9, P. 439-449.
Journal of Engineering Volume 13 September2006 Number 3
8361
SLANTLET TRANSFORM-BASED OFDM SCHEME
Saifuldeen A.Mohammed
Mohammed Nadhim Abbas Sulaiman M.Abbas
University of Baghdad,
Baghdad,Iraq
Department of Electrical
Engineering
Ass.Prof Department of Electrical
Engineering
الخلاصة
مما يستدعي الحاجة إلى أنظمة موثوقة وذات كفاءة طيفيةة عاليةة و لتحقية ة اللاسلكية في توسع مستمر الرقميإن الاتصالات
استحوذت على الكثير من الاهتمام. في هذا البحة اقترتةط طريقةة جديةدة (OFDM)هذه الحاجة فأن تقنية مزج الترددات المتعامدة
ا البح هو تقليل مستوى التةدالل وااالةة الحاجةة الةى اسةتعمال الفتةرة الفا ةلة هذ (. ان الطريقة الجديدة فيOFDMلتحسين اداء ال)
,هةةذا يةةتس عةةن طريةة تبةةديل تحويةةل الفةةو يير بطريقةةة (OFDM( فةةي منمةةومة ال)Bandwidthوبهةةذا تةةس تحسةةين كفةةاءة النطةةا )
( افمةل SLT-OFDMقةد لةوتا ان )ف (Wavelet)( هو تطويرتصل في المويجةات Slantlet. وان ال )(Slantletالأنحدا المائل)
( اسةةةرك بكثيةةةر مةةةن لوا اميةةةة Slantletفةةةي تالةةةة القنةةةاة الةةةوهن الانتقةةةائي للتةةةردد و كةةةذل ان لوا اميةةةة ) (WP-OFDM)مةةةن
(Wavelet( في هذا البح تس التركيز علةى المقا نةة بةين .)FFT-OFDMو ) (SLT-OFDM مةن اهةس النتةائل التةي تةس الحصةول )
المسةةتو . بينمةةا اداء الةةوهن ة( فةةي قنةةاFFT-OFDMعةةن اداء )dB 18يكةةون افمةةل بحةةدود (SLT-OFDM)عليهةةا هةةو ان اداء
فةةي تالةةة القنةةاة الةةوهن SNRالواطئةةة وسةةوع يةةنعكء الاداء مةةع ايةةادة SNRيكةةون افمةةل فقةةن فةةي منطقةةة (SLT-OFDM)نظةةام
الانتقائي للتردد.
ABSTRACT
Wireless digital communication is rapidly expanding resulting in a demand for systems that
are reliable and have a high spectral efficiency. To fulfill these demands OFDM technology has drawn
a lot of attention.
In this paper a new technique is proposed to improve the performance of OFDM. The new technique
is use the slantlet transform (SLT) instead Fast Fourier transform (FFT) in order to reduce the level of
interference. This also will remove the need for Guard interval (GI) in the case of the FFT-OFDM and
therefore improve the bandwidth efficiency of the OFDM. The SLT-OFDM is also better than wavelet
packet (WP)-OFDM in the selective channel because the slantlet filter bank is less frequency selective
than the traditional DWT filter bank, due to the shorter length of the filters and SLT algorithm is faster
than WP algorithm. The main results obtained indicate that the performance of SLT-OFDM is better
on average by 18dB in comparison with that of FFT-OFDM flat fading channels. For frequency
selective fading channel the SLT-OFDM performs is better than the FFT-OFDM on the lower SNR
region, while the situation will reverse with increase SNR values.
WORD KEY
slant let transform - OFDM- Guard interval
S. M.Abbass Slantlet Trans form-Based OFDM Scheme
M. N.Abbass
S.A.Mohammed
8361
INTRODUCTION
Orthogonal Frequency Division Multiplexing (OFDM) is very similar to the well known and
used technique of Frequency Division Multiplexing (FDM). OFDM uses the principles of FDM to
allow multiple messages to be sent over a single radio channel. It is however in a much more
controlled manner, allowing an improved spectral efficiency.
The Fourier transform (or other transform) data communication system is a realization of FDM in
which discrete Fourier transform are computed as part of modulation and demodulation process. In
addition to eliminating the banks of subcarrier oscillators and coherent demodulators usually required
in FDM system, a completely digital implementation can be built around a special-purpose computer
performing the fast Fourier transform [1]. OFDM has recently been applied widely in wireless
communication systems due to its high data rate transmission capability with high bandwidth
efficiency and its robustness to multi-path delay. It has been used in wireless LAN standards such as
American IEEE802.11a and the European equivalent HIPERLAN/2 and in multimedia wireless
services such as Japanese Multimedia Mobile Access Communications. A dynamic estimation of
channel is necessary before the demodulation of OFDM signals since the radio channel is frequency
selective and time-varying for wideband mobile communication systems [2].
Recently, Selesnick has constructed the new orthogonal discrete wavelet transform, called the
slantlet wavelet, with two zero moments and with improved time localization [3]. This Transform
method have played an important role in signal and image processing applications. The slantlet has
been successfully applied in compression and denoising. It also retains the basic characteristic of the
usual filterbank such as octave band characteristic, a scale dilation factor of two and efficient
implementation. However, the SLT is based on the principle of designing different filters for different
scales unlike iterated filterbank approaches for the DWT [4].
SLANT LET FILTER BANK [5]
It is useful to consider first the usual iterated DWT filter bank and an equivalent form, shown in
Figure 1. The symbol a1 is the symbol with the highest frequency, while symbol a4 is the symbol with
the lowest frequency. The ‘slantlet’ filter bank described here is based on the second structure in
figure (1.b), but it will be occupied by different filters, that are not products. With the extra degrees of
freedom obtained by giving up the product form, it is possible to design filters of shorter length, while
satisfying orthogonality and zero moment conditions, as will be shown. For the two-channel case, the
shortest filters for which the filter bank is orthogonal and having K zero moments, are the well known
filters described by Daubechies [6]. For K = 2 zero moments, those filters H ( z ) and F ( z ) are of
length 4. For this system, designated D2,the iterated filters in Figure 1 are of length 10 and 4. Without
the constraint that the filters are products, an orthogonal filter bank with K = 2 zero moments can be
obtained where the filter lengths are 8 and 4, as shown in Figure 2, side by side with the iterated D2
system. That reduction of two samples grows with the number of stages, as in Figure 3. We make
several comments regarding Figures 2 and 3.
- Each filter bank (equivalently, discrete-time basis) is orthogonal. The filters in the synthesis filter
bank are obtained by time-reversal of the analysis filters.
- Each filter bank has 2 zero moments. The filters (except for the lowpass ones) annihilate
discrete-time polynomials of degree less than 2.
- Each filter bank has an octave-band characteristic.
- The scale-dilation factor is 2 for each filter bank. Between scales, the filters dilate by roughly a
factor of 2. (In the slantlet filter banks, by exactly a factor of 2.)
- Each filter bank provides a multiresolution decomposition. By discarding the highpass channels,
and passing only the lowpass channel outputs through the synthesis filter bank, a lower resolution
version of the original signal is obtained.
Journal of Engineering Volume 13 September2006 Number 3
8361
- The slantlet filter bank is less frequency selective than the traditional DWT filter bank, due to the
shorter length of the filters. The time-localization is improved with a degradation of frequency
selectivity.
- The slantlet filters are piecewise linear.
- In figure 1 it is clear that DWT needs two stages while Slantlet needs one stage only.
It must be admitted that, although both types of filter banks posses the same number of zero
moments, the smoothness properties of the filters are somewhat different. In Figures 2 and 3, the
slantlet filters have greater "jumps" than do the iterated D2 filters.
However, the Haar basis, with its discontinuities, is suitable for analyzing piecewise constant
functions that have jumps. Likewise, the slantlet filter bank appears appropriate for the analysis of
piecewise linear functions, as illustrated in the denoising example below.
The ability to model jumps is also relevant for other applications, like edge detection and
change point analysis, in which the detection of abrupt changes in an otherwise relatively smooth but
unknown function is considered [7]. In figure(2) the S1 is frequency response of F(z) and S2 is
frequency response of H(z)F(z2) and S3 is frequency response of H(z)H(z
2) and V1 is frequency
response of G1(z) and V2 is frequency response of F2(Z) and V3 is frequency response of H2(Z) and
also in figure(3) the S1 is frequency response of F(z) and S2 is frequency response of H(z)F(z2) and
S3 is frequency response of H(z)H(z2)F(z
4) and S4 is frequency response of H(z)H(z
2)H(z
4) and V1 is
frequency response of G1(z) and V2 is frequency response of G2(z) and V3 is frequency response of
F3(z) and V4 is frequency response of H3(z).
Stage 1 Stage 2
a1
a2
Input signal x(n)
a3
a4
(a) DWT filter bank
a1
Input signal x(n) a2
a3
.
a4
(b) slantlet filter bank Fig. 1: Iterated filter bank and an equivalent structure.
S. M.Abbass Slantlet Trans form-Based OFDM Scheme
M. N.Abbass
S.A.Mohammed
8368
(a) Block diagram
(a) Block diagram
(b) Impulse response of filter bank.
(b) Impulse response of filter bank .
(c) Frequency response.
Fig. 3: Comparison of iterated D2 filter
bank (left-hand side) and slantlet filter
bank (right-hand side).
(c) Frequency response.
Fig. 2: Comparison of iterated D2 filter
bank (left-hand side) and slantlet filter
bank (right-hand side).
Journal of Engineering Volume 13 September2006 Number 3
8361
n(t)
A SYSTEM FOR FFT-BASED OFDM
The block diagram of the system for OFDM is depicted in figure (4).
I/P
DATA
TRANSMITTER
CHANNEL
O/P
DATA
RECEIVER
The OFDM modulator and demodulator of FFT-based OFDM is shown in figure (5).
Fig. 4: Block Diagram of OFDM System.
Serial
To
Parallel
Signal
Mapper
Adaptation
of
Transmission
Frame
Parallel
To
Serial
Generation
of
Pilot Carriers
OFDM
Modulato
r
Parallel
To
Serial
Serial
To
Parallel
Channel
Compens
ator
Channel
Estimator
OFDM
Demodulat
or
Signal
Demappe
r
+ Multipath
Rayleigh
channel
S. M.Abbass Slantlet Trans form-Based OFDM Scheme
M. N.Abbass
S.A.Mohammed
8366
(a) OFDM Modulator.
(b) OFDM Demodulator.
Fig. 5: The OFDM modem system.
First of all, the input serial data stream is formatted into the word size required for
transmission e.g. 2 bit/word for QPSK and 4 bit/word for 16-QAM, and shift into a parallel format.
The data is then transmitted in parallel by assigning each word to one sub-carrier in the
transmission. After that, the data to be transmitted on each sub-carrier is then mapped into QPSK or
16-QAM constellation format. This process will convert data to corresponding value of M-ary
constellation which is complex word, i.e. real and imaginary part. The training frame (pilot sub-
carriers frame) will be inserted and sent prior to information frame. This pilot frame will be used for
channel estimation that's used to compensate the channel effects on the signal. After that, the
complex words frame and pilots frame will pass to IFFT to generate an OFDM symbol. Zeros will
be inserted in some bins of the IFFT in order to make the transmitted spectrum compacts and reduce
the adjacent carriers interference.
PROPOSED SYSTEM FOR SLANTLET TRANSFORM -OFDM
The overall system of OFDM is the same as in figure (4). The only difference is the OFDM
modulator and demodulator. The slantlet transform SLT-OFDM modulator and demodulator that
used are shown in the figure below:
Journal of Engineering Volume 13 September2006 Number 3
8366
(a) SLT-OFDM modulator.
(b)ISLT-OFDM demodulator.
Fig. 6: SLT-OFDM modem system
The processes of the S/P converter, the signal demapper and the insertion of training
sequence are the same as in the system of FFT-OFDM. Also the zeros will be added as in the FFT
based case and for the same reasons. After that the inverse slantlet transform (ISLT) will be applied
to the signal.
The main and important difference between FFT based OFDM and SLT based OFDM is
that the SLT based OFDM will not add a cyclic prefix to OFDM symbol. Therefore the data rates in
SLT based OFDM can surpass those of the FFT implementation. After that the P/S converter will
convert the OFDM symbol to its serial version and will be sent through the channel.
At the receiver, also assuming synchronization conditions are satisfied, first S/P converts the
OFDM symbol to parallel version. After that the SLT will be done. Also the zero pad will remove
and the other operations of the channel estimation, channel compensation, signal demapper and P/S
will be performed in a similar manner to that of the FFT based OFDM.
PERFORMANCE OF THE OFDM SYSTEMS IN THE FLAT FADING CHANNEL:
In this type of channel, the signal will be affected by the flat fading with addition to AWGN
(Additive White Gaussian Noise), in this case all the frequency components in the signal will be
affected by a constant attenuation and linear phase distortion of the channel, which has been chosen
to have a Rayleigh's distribution. A Doppler frequency (Doppler Shift) of 50 & 100 Hz is used in
this simulation and the table (1) explains the simulation parameters. Figure (7) show the BER
performance of SLT-OFDM and FFT-OFDM in flat fading channel for QPSK modulation type.
Table (1): Simulation Parameters.
64 Number of sub-carriers
64 Number of SLT points
64 Number of FFT points
Flat fading+AWGN
Channel model Frequency selective fading+AWGN
S. M.Abbass Slantlet Trans form-Based OFDM Scheme
M. N.Abbass
S.A.Mohammed
8361
0 5 10 15 20 25 30 35 4010
-5
10-4
10-3
10-2
10-1
100
BER Vs SNR
SNR[dB]
BE
R
SLT-OFDM-100Doppler shift
SLT-OFDM-50Doppler shift
FFT-OFDM-100Doppler shift
FFT-OFDM-50Doppler shift
Fig. 7: BER performance of SLT and FFT- OFDM for QPSK modulation in flat fading
channel.
The Performance of OFDM Systems in Frequency Selective Fading Channel (multipart's-
channel).
The BER performance of SLT and FFT-OFDM systems in frequency selective fading channel
are shown in figure (8). This case corresponding to multipaths where two paths are chosen and the
attenuation and delay of the second path are -8dB and 8 samples respectively. From the figure (8), it
is clear that the BER performance of SLT-OFDM will become constant after a certain SNR. From
the same figure, one can see that the BER curves of FFT-OFDM will decrease with the increase of
the SNR. In the frequency selective fading the SLT-OFDM is not better than the FFT-OFDM for all
the SNR.
Journal of Engineering Volume 13 September2006 Number 3
8363
0 5 10 15 20 25 30 35 4010
-5
10-4
10-3
10-2
10-1
100
BER Vs SNR
SNR[dB]
BE
R
SLT-OFDM-100Doppler shift
SLT-OFDM-50Doppler shift
FFT-OFDM-100Doppler shift
FFT-OFDM-50Doppler shift
Fig. 8: BER performance of SLT and FFT-OFDM for QPSK modulation in selective fading
channel.
The above results can be interpret as follows. The FFT-OFDM has a guard interval(cyclic prefix)
of 25% this mean that a cyclic prefix is equivalent to 16-samples, therefore no ISI will effect on the
FFT-OFDM until the delay of the second path exceed 16 samples. Since the delay of the second
path is equal to 8-samples as assumed above, no ISI will effect on it, while in SLT-OFDM there's
no cyclic prefix this mean that ISI will occur in SLT-OFDM. Also due to high spectral containment
between the sub-channels in SLT,SLT-OFDM will robust again ISI and ICI until a certain SNR
value, after this value , the SLT-OFDM performance will be constant approximately with the
increasing of SNR and the FFT-OFDM performance will become better than it.
CONCLUSION
In flat fading channel, it was found that the SLT-OFDM performance was better than that of
the FFT-OFDM. A gain of about 18dB was obtained in SLT-OFDM over that for the FFT-OFDM
and also the effect of Doppler Shift is very slightly in SLT-OFDM. But in frequency selective
fading channel (multipaths case), the situation will be changed. Since the Cyclic Prefix (CP) which
is already exists in the FFT-OFDM will eliminate the ISI, therefore no ISI will occurred in FFT-
OFDM if the CP is greater than the delay spread of multipaths (in this case we considered that this
condition is satisfied ). In the case of WP-OFDM there's no CP therefore ISI will occurred.
Therefore the BER performance of SLT-OFDM was better than the FFT-OFDM case until a certain
value of SNR. After this value the FFT-OFDM was better than SLT-OFDM. It was noticed that the
BER curves of SLT-OFDM will become flat (constant with the increase of SNR).
S. M.Abbass Slantlet Trans form-Based OFDM Scheme
M. N.Abbass
S.A.Mohammed
8361
REFERENCES
[1] S. B. Weinstein and Paul M. Ebert, “Data Transmission by Frequency-Division Multiplexing
Using the Discrete Fourier Transform”, IEEE Transactions on Communication Technology,
Vol. COM-19, No. 5, October 1971, pp. 628 – 634
[2] A.Bahai, S.Coleri, M.Ergen, A.Puri"Channel Estimation Techniques Based on Pilot
Arrangement in OFDM systems ", IEEE Trans. on Broadcasting, Vol.48, No.3, pp 223-229,
September 2002.
[3] Edward R. Dougherty, Jaakko T. Astola, Karen O. Egiazarian "The fast parametric slantlet
transform with applications" Proceedings of SPIE -- Volume 5298, May 2004, pp. 1-12.
[4] G. Panda, P. K. Dash, A. K. Pradhan, and S. K. Meher "Data Compression of Power Quality
Events Using the Slantlet Transform" IEEE TRANSACTIONS ON POWER DELIVERY,
VOL. 17, NO. 2, APRIL 2002.
[5] Ivan W. Selesnick" THE SLANTLET TRANSFORM" Polytechnic University Electrical Engineering 6
Metrotech Center, Brooklyn, 0-7803-5073 1998 IEEE
[6] I. Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, PA, first edition, 1992.
[7] T. Ogden and E. Parzen. Data dependent wavelet thresholding in nonparametric regression with
change-point applications. Computational Statistics and Data Analysis, 2253-70, 1996.
3العدد 7002 ايلول 33المجلد الهندسة مجلة
-181-
طرائق الازالة في تقليل اعداد الطحالب في مياه الاحواض المكشوفة
Removal Method Assessment Through Reduction Algae Count in
Uncovered water Reservoir
ثائر ابراهيم قاسمد.
Dr. Thaer I. Kassim رافع هاشم السهيليأ.د
Prof. Dr. Rafa H. Suhaili
ادق سلمانحمد صم . م مMohammed S. Salman
كلية الهندسة -جامعة بغداد كلية الهندسة –جامعة بغداد معهد الهندسة الوراثية -جامعة بغداد
الخلاصــة
تعددا طحالبحدد لددب طحلاددئئبي طحي ماددم اددة ليددبطا طاددع اددة للاددبي طحلمددبل طات ددبي طحئلمدديطي طح تدد طي اة طحاعم طحيط ل أتااطا طح ت طي تغمدي صادب ق عبيامد طحلمدبل تتمتد حت طتدا ب طاي ط م ، لب تائئه لب ليبطا
طيتلددبا ايط ددس طحاماددديح طحطملمب مدد اددة ددلط طحئلددت تددم ئطثدديح لمددت أيتئدديي طحالبحدد لعمددبيطم حت دد ت طحلمددبل طحاددالم ب مدد لثددا طحيدد ئتيططمددا ( لحددم لددب صددخا طاددتصاطم ليطئددبي طملم(Jar Testلصتئيمددب ئباددتصاطم ت ددبا الددق طحتدديح
ئبحتدددبحة ت مدددا طيددداطا طحالبحددد 94%)( ل غم/حتدددي تدددم طحلاددد ا ي ددد تادددئ ت مدددا طحعطددد يح ئل ددداطي )10 - 50)%( طضددبا طحدد لحددم تددم طاددتصاطم ليطئددبي طملمب مدد لثددا ئيلتطتددبي طحئ تبادددم م طئيمتددبي طحتلدددب لددباح 95ئتاددئ )
طضددبا طحتيطمددا طالثددا ح يدد طبتددي تتددب تادد ت مددا ئعددا ( ل غم/حتددي 1 - 5طح ددبمئ ط يطمب )ط دد ي( ئتيططمددا لددب ) ي طحت طحة %99 %100 %99.2ي طحت طحة تا أاطح طحالبح %95 %79.3 %95طحعط يح
Abstract The Algae is considered as one of the major causes of some serious problems that occur in
water plants, rivers, lakes and irrigation channels. Those problems are the unpleasant taste and odor,
the clogging of waterways, and others. Hence, the algae existing extensively in any water body is
considered as an obvious indication of surface water pollution. Chemical control methods were used in this research for reducing the turbidity and Algae in the
laboratory using the (Jar Test). This was done by using chemical materials like Alum with
concentration (10 - 50 mg/l). The percentage of the reduction in the algae was (95%) and in turbidity
(94%). It is shown also that when using KMnO4, CuSO4 and Cl, each separately after adding the ideal
dose of alum found before, will reduce the turbidity with ( 95% ,79.3% .95%) and algae removal of
(99.2%, 100%, 99%) respectively
182
، لعبحتد ، طملمب مد ، امامب مد ، طثديطا يدلط ة ، طحالبح ، طااطح ، ت مدا ، طحعطد يح ، ادمايح :الكلمات الرئيسية .ح تم ئبم
المقدمة Drainageتائ طحالبحد ظ ي ياا لب طحليبطا اة طحلمبل لمت تعمس ليطد طحلدبا ادة أتظلد طحت يمد
System ، تؤثي ي تل طاالبم ليطت ب ئ م طايطبا طحلمبتم اة طحلبا طلإتااطا ئبحلضصبي طحالبلبي تاددئ طددلحم ليددط طحيط لدد طحاعددم اددة طحلمددبل للددب مددؤال طحدد لصددباي اددلم ي دد طحلمددبح طحئيدديم طحلمددبح طحلب مدد
(Rashash et al.,1996) يددب طحتيددبا طحئيدديل طحعل مددبي طااامددبا ئاددئ طيت ددب تاددئ طحلغددلمبي طحتبتتدد لبحدد ددلل طحليددبطا تئددا اددة
طحائمعم ، ئت طح عي تبم ا يبحلة طضح ل ا الق ت يم طحلبا طحلاتصام للب يا لطيل تبم لتا أ طحامايح ي تل طحالبح لأائب ئم م الم ئييم لب طضح طحلا
امايح ي م ب ئبحايط س طحتة تادتصام حئ مد طحالبح اغميح طحلتم ايمع طحتل ئباضبا إح أته لب طحاع طح ئدبحايس طحلمطبتمطمد طحتدة تادبيا ادة ت مدا طثدي دلل طحليدبطا لدأ طحالبحد طحصمامد لطحتئبتبي طحلب م لثدا طح ادأ أ
أب طحد متأثي تل ئعض طحالبح ئأاتصاطم طحلئماطي حطب تبم تأثميطي تبتئم ي د طحادل طحعبلد طحئم د إضدبا اطي تددؤثي ي دد تلمددأ طتدد ط طلألمددبا اددة طحلاددالبي طحلب مدد ئبلإضددبا إحدد أب طحالبحدد تعدد ا لدديح طصددي ددلل طحلئمدد
ح تل ئعا ا طا تأثمي طحلئماطي )إتص بض تيطمال( عا مط ب طحتل ئيطا أطثف لب طحابئس -طرائق مستخدمة في عملية السيطرةعلى الطحالب منها:ال الغذائي السيطرة على عملية الاثراء -أ
طحلغددلمبي لاددباي تعددا ايط ددس طحاددمايح ي دد يل مدد طاثدديطا طحغددلط ة ددة طلأ ددم ، م اددا ئ ددب طحاددمايح ي دد Kathandaraman and)لمدت ططدا ،لمت متم طحامايح ي طحلتبئأ طحتة مط ب طاثيطا طحغلط ة ام ب ئتا يبحمد
Evans, 1983) خا طااطيح طحتماح حلاباي طحلمبل ادة طحئلمديطي طب طحامايح ي طحالبح اة طحئلميطي تتم لب ص اعددد طحعل مدددبي طحايطيمددد طحادددمايح طحتبلددد ي ددد يل مددد لعبحتددد طحددد طحددد ضدددي يح طحلبتددد (Sawyer,1968)بي دطيددد
Makenthun and) تددبع طحلص ددبي طحلاتمدد طحاددب طحلص ددبي طحاددتبيم ح لددا لددب ظددب يح طاثدديطا طحغددلط ة
kemp,1970) ام لثبحمدد طحت تمدد طحلاددتصال لأاطحدد طحلغددلمبي ئ اددبا طاددتصاطم طاددلبم ط تلدد طحتئبتددبي أ أمضددبم يدد طلطبتم تليمم طحي ططا طح بيم ئ ابا ل طا يمي عبئ ح ت بيا
3العدد 7002 ايلول 33المجلد الهندسة مجلة
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السيطرة الفيزيائية –ب
طاددئب أ طاددتصاطم Harvestingتتضددلب ايط ددس طحاددمايح طح مامب مدد ي دد طحالبحدد طااطحدد طحلمطبتمطمدد ح ددب طحتدة تدؤثي ي د ص ص د (Ultrasonic)حغيض ت مدا ت د ل طحضد ا طاطحد طحي ططدا طدلحم طادتصاطم يدلتبي ط أيدع
طحغددبا اددة طحاددب ا طحص دد ل ح البحدد ي دد طالمددبا طاصددي طمضددبم ددلط يمددي ل ئدد ا ، أ طحاددمايح ي م ددب لددب صددخا طحلتيدا حطدب حددم مطدب تددبم طعئدبا طضدح ح ددلل طحايم د اددة طحدتلطم ئلادت طحلددبا تيمبتده طدلحم تددم طادتصاطم طح لددم
Powdered) ب طحلتيدددا دطحطبيئددد لادددل ساطم ع مدددا لدددب دملطدددب إادتصددد أل (Janik,1980)صاطتدددبي تت مدددا طحلمدددبل
activated carbon) مابيا ي تتلأ أ تصثدي طحالبحد ئبحتدبحة ملطدب طحلل طحلباح امايح ي طحالبح اة (Morris and Riley, 1963)إاطحت ب
السيطرة الكيميائية -جـ لب طحايط س طاصي اة طحادمايح ي د طحالبحدد دة طحامادديح طحطملمب مدد ئباتصدداطم طحلئمداطي لدب طحليطئدبي
ئيلتطتددبي (Steel and McGhee , 1979)طحيددب ع طااتصددداطم ددة ليطئددبي طحتلددب لثددا طئيمتددبي طحتلددب اطم دا اح طلإاتصدد ألب طحلئماطي طاصي ا ة للد(Ficek, 1983)ئتتبح اة ئعض طحلباي طحئ تبام م طحلل طاتصام
ص دما لددب طئيمتددبي طحتلدددب لددأ Rosin amines Triazine derivativeا د لثدديح ي د طحالبحدددة طحامادددادا أصتئددددبي أطثددددي لددددب دة أللدددبض يضددد م أحام بمددداطي طمت تددددبي ئعدددد م طحيئبيددددبي طال تمددددتتدددديطي طح ضددد ليطئددد
ا طحليطئددبي د أاضددد دد P-chlorphenyl-2thienyl iodonium chlorideليطددد يضددد ل ثئدددي ئددأب 10000 ( (Prows and McIlhenny, 1974يح ي طحالبح داعبحم اة طحاما
ح ط دددد يلدددا طثمدددي لدددب طحئددددبلثمب ئأتتدددبل طمتدددبا ئدددداط ا يدددب طئيمتدددبي طحتلددددب ح ادددمايح ي ددد طحالبحدددد حطدددب تظددديطم طحدلل معتئدي لد طا Alumمعتئي طحط د ي طحيد طاعتابام طح طا حطئيمتبي طحتلب تع ت ب ل ض اة طااتصاطم ،
ئبحييم لب طحليبطا طحالم تتبل طحلمدبح طحلب مد تاتصام لب طتا طحامايح ي طحالبح لب أاضا طحل طا طحتة لصثيح (Sawyer, 1968)ل طا طحتة أصلي ئتظي طلأيتئبياة إاتصاطم لل طح
السيطرة البايولوجية – د ملطدب طادتصاطم اطديح تظدبم طحادمايح طحئبم ح تمد لدأ طلأصدل ئتظدي طايتئدبي أت دب تادئ ضديي ي د طحئم د ئداام لددب طحامايح ئبحايط س طحطملمب مد اطثمدي لدب طحايط دس طحئبم ح تمد طحلتت يد ح ادمايح ي د طحالبحد تدم طئتطبي دب تتضدلب
اتصاطم لتبلمأ تبيا ح طيف لثا طائتاط مبي طح ب لبي طحلم طتم طلأالبم إا تصاطم ابمي ابي ئطتيمب اايمبي ط (Dunst, 1974)
الجانب العملي
184
تدددم طتددديطا تتدددبي حايطاددد تدددأثمي ئعدددض طحليطئدددبي طحطملمب مددد لثدددا طئيمتدددبي طحتلدددب ئيلتطتدددبي طحئ تبادددم م ئ تد ا طحيد طحدلل تدم Raw Water حلألد طضإاطحت دب لدب لاداي طحلمدبل طحاطص د طحط د يطمب ي د طحالبحد يل مد
Coagulantطاتصاطله طلصثي أثددي طحتيطمدا طح عددبا ايطاد طحغدديض لدب ددلل طحتتدبي دد Jar Test تلدي طحتتدبي طيددخل ئبادتصاطم ت ددبا
ح ل طا طحلضبا اة يل م إاطح طحالبح ت ما طحعط يح ئيل تلضمي طحللبحما طحخال حطا تتيئ طلب م ة :تضلب طحعلا طحلصت
محلول الشب -أ
( (Al2SO4.18H2Oيدديطم لددب لئمئددبي ط لاددل س طحيدد 1دي لل دد ا طحيددد لددب صدددخا طلطئددد مددلضتددم تلا لب طحلدبا طحل ادي تلي يل م طحلداج 100 طحلل مالد لل مدبم ئيد طحللا طحلت طاي اة للابي طحتا م اة
حلاح ابي لمت متم طحلا ا ي لل د ا لدب طحيد تيطمدال Magnetic Stirrerد ت دبا ا طي لغتباماة ئ اباحتدي 1 ئبحتائدد ح لب مددبي اعدد Jar Test% لب لط طحلل دد ا تدم تلضدمي للبحمددا طحتتيئدد حلعبل ت دب ئت دبا1
طحت طحة لدب طحلل د ا مضدبف طحد صلد لا ي 5 4 3 2 1 تلتد ل ي لمبل طحتل لج لمت متم ال ( 10, 20, 30, 40, 50لب مدبي لدب طادا ادت حغديض طحلاد ا ي د تيطمدا طحيد ادة دلل طحلب مدبي ئل داطي )
اددس طحصادد طي طحتبحمدد ئعددا طضددبا طحيدد طحدد طحلب مددبي Jar Testاطم ت ددبا د مددتم أاتصدد ل غم/حتددي ي دد طحتدد طحة طضبا طحصلا لمت تتيم طحلب م طحاباا ئخ
ا يح ئبحاعم حغيض تتبت طحلل ا اة طحلب مبي 200طتيطا لاج ايمأ حلاح اعم طلاح ئايي -أ اعم 20ا يح ئبحاعم حلاح 20طتيطا لاج ئائ ئايي - اعم حغيض طحتيطما 30طم بف يل م طحلاج طحئاة تيم طحللبحما حلاح -تد أاضا تيي ح ي طحتة تعاة أ ائ عيطاح ح عط يح بالبح حغيض ئمب أياطا طح pHعمب طحعط يح -ا
والكلوراين وكبريتات النحاس.محلول برمنكنات البوتاسيوم -ب يديطم لدب 1لدب صدخا طلطئد تيأ أتط مدالل99.5 % ل ت ب ح KMnO4 ا ئيلتطتدبي طحئ تبام م دلضي لل
0.3 0.2 0.1صددل لتده % طحدلل أ1لل دد ا ئتيطمدددا لدا لدب طحلدبا طحل ادي ح لادد ا ي د 100طحلدباح ادة ت طحة حطة متم طضباته طح تلدبلج طحلدبا طحتدة تلتد ل لادئ ب ي د طحتييد طحلثبحمد ح يد لدب لا ي طح 0.5 0.4
م ل غد ( 1 , 2 , 3 ,4 , 5)طحصا ح طحابئ حلحم ادمط ب تيططمدا لل د ا ئيلتطتدبي طحئ تبادم م ادة صلد لب مدبي ددة طحالبح أياطا pH متم عمب طحعط يح Jar Test/حتي متم طيباح طحصاد طي أ ، ، تد حعل م طاتصاطم ت دبا
طددلحم طحلددبا اددة يل مدد تلضددمي طاضددا تييدد حئيلتطتددبي طحئ تباددم م طحتددة تعاددة طاضددا طح دديطاطي ض ئمددببحغددي لل ا طحط يطمب طئيمتبي طحتلب
3العدد 7002 ايلول 33المجلد الهندسة مجلة
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النتائج والمناقشةتلتددبا ددلل ( لمددت (Jar Testب طحايط ددس طحت مامدد حت ددمم يل مدد طحتصثمددي طااطحدد ددة الددق طحتدديح أب لدد
طحايم ئبحئابا طحلي ت تعاة ا يح طضدل يدب طحتصثمدي طحت ئمدا طحتيادم لع لدبي ل اد يدب اتديح طحت ئمدا لمطبتمطم طحلاج تأثمي ئعض طحع طلا لثا ايت طحليطيح طلم ت يم طحلصثدي طا طح مداي تمتة ، حدل تعتئدي أ دم
تل لج لاغي حت ممم طدا طحلتغمديطي ادة Jar Testحلصثي معتئي أا ام لصتئيم اة أمتبا أتا تيي لب ط (Hannah et al., 1967)يل م طحتصثمي يتالب تط ب لاطي طحلعبحت يمي ل ت اح
0
20
40
60
80
100
0 10 20 30 40 50تركيز الشب ملغم/لتر
رةكو
لع الة
زا ابة
سن
0
20
40
60
80
100
120
0 10 20 30 40 50تركيز الشب ملغم/لتر
بحال
لطة ا
زالة ا
سبن
مع تراكيز للعكورة( تغيرات نسب الازالة 3شكل )
مدة البحثالشب خلال مع تراكيز الطحالب( تغيرات نسب الازالة 7شكل )
مدة البحثخلال الشب
استخدام الشب
ئددأب لددباح طحيدد ح ددب (7والشــكل 3الشــكل ) متضددح لددب صددخا طحتتددب طحلاتلادد لددب طحتتددبي طحلصتئيمدد % لددب طحعطدد يح طحط مدد ، ئبحتددبحة ت مددا طيدداطا طحالبحدد لددب 30.4 – 94.6 طلطبتمددد طاطحددد ئتاددئ تتدديط ح ئددمب
- 40% طي د تيططمددا طااتد ح يدد أتلاديي ئدمب 95تادئ طاطحد ايا/حتدي طل ئ 6197ايا/حتدي طحد 2708253
ل غم /حتي 30أضددبا طحلددلحم اددأب عملدد طا طح مدداي تمتة عددا تتبعاددي ئامددباح تيطمددا طحيدد طحلدد طا طحطملمب مدد طاصددي لددب
متة حتل مددس أل تدا ئدأب أاضدا عملدد حدخ طح مداي ت (Kim ,2000) لط لب متابئس لدأ لدب الظده (8.1 – 7.2)يل مدد طحتصثمددي تعتلدددا ي دد طح ملددد طا حمدد ح بيامدد ، لمددت أتدده ئامددباح عبيامدد طحلل دد ا اددأب لددا طا طح مدداي تمتة
(1طلب اة تا ا يعم )ا ف م ا
( قيم تغير الاس الهيدروجيني مع أختلاف تراكيز المواد الكيمياوية بثبوت الشب3) تا ا
pH تراكيزCuSO4 mg/l pH يز تراكCl2
Mg/l pH تراكيزKMnO4 mg/l pH تراكيز الشب
mg/l
186
8.1 Raw Water 8.1 Raw Water 8.1 Raw Water 8.1 Raw Water 7.7 0 7.9 0 7.3 0 8.1 0 7.2 1 7.6 1 7.3 1 7.7 10 7.2 7 7.4 7 7.3 7 2.7 70
7.2 3 7.3 3 7.3 3 2.3 30
7.2 4 7.3 4 7.3 4 2.3 40
7.2 7 7.2 7 7.3 7 2.7 70
استخدام برمنكنات البوتاسيوم
70758085
9095
100
0 1 2 3 4 5
تركيز برمنكنات البوتاسيوم ملغم/لتر
رةكو
لع الة
زا ابة
سن
90
92
94
96
98
100
0 1 2 3 4 5
تركيز برمنكنات البوتاسيوم ملغم/لتر
بالحط ال
لةزا
ابة
سن
مع تراكيز للعكورة( تغيرات نسب الازالة 3شكل )
مدة البحثخلال برمنكنات البوتاسيوم بثبوت الشبمع تراكيز الطحالب( تغيرات نسب الازالة 4شكل )
مدة البحثخلال برمنكنات البوتاسيوم بثبوت الشب
طيتلبا طحتيططما طاات ح ي لمدت ام م طلؤطاا طئتاط ة طحط يح طائتاط مطلب يتا طاتصاطم ئيلتطبي طحئ تبملادب ط دباح طاطا يل مد (4و الشـكل 3الشـكل ) أتضح ئبب طاتصاطم ئيلتطبي طحئ تبام م لأ طحتيطما طاات ح يد
اديا / حتدي 2708253 ئبحتدبحة ت مدا طيداطا طحالبحدد لدب % 98% طحد 88طحتصثمي اة إاطح طحعط يح ئتائ لب % معدددا طحاددئ اددة لحددم طحدد تطددد ب ثتدددب ة ط طادددما طحلتغتمدددا 96.2 – 99.5طحددد طحادد ي ئتاددئ طاطحدد عدداي ب
(Manganese dioxide صددخا يل مدد طاطادداح ئبحئيلتطددبي طحددلل مددؤال طحدد تلاددمب ط ددباح طحتصثمددي ئبحتددبحة )ما ل ئداطي تدؤال طحد طحتيادم ط عدا تعدا طمضدب طحد عبئ مد أ طادما تطامي طحل طا طحعض م طحغي م لب صدخا تيدط
طحلتغتمددا طحعبحمدد ي دد طالااددبق ح لدد طا طحعضدد م لددب صددخا طا طاددي طح ماي تمتمدد ئبحتددبحة تطدد ب ل ئدداطي طئمدديح (Ma et al., 1997)طحلتم طحتة تاما لب يل م طحتيام ئبحتبلئم
3العدد 7002 ايلول 33المجلد الهندسة مجلة
-187-
استخدام مادة الكلور
0
20
40
60
80
100
120
0 1 2 3 4 5
تركيز الكلوراين ملغم/لتر
رةكو
لع الة
زا ابة
سن
0
20
40
60
80
100
0 1 2 3 4 5
تركيز الكلوراين ملغم/لتر
بحال
ط ال
لةزا
ة اسب
ن
بثبوت الكلورمع تراكيز للعكورة( تغيرات نسب الازالة 7شكل )
مدة البحثخلال الشبالكلور مع تراكيز الطحالب( تغيرات نسب الازالة 6شكل )
مدة البحثخلال بثبوت الشب
ي طلب يتا طاتصاطم تيططما طحط يح طائتاط م ئبيتلبا طحتيطما طاات ح ي ا ا ح لظ ئبب طحتيططما طح م ح ط% حطددب يتددا طاددتصاطم طحتيططمددا طلأي دد مددؤال طحدد 95ت ددا طحعطدد يح ح لمددبل طحلتياددئ ئتاددئ ( 6 طحيددطا 5)طحيددطا
( % 93 - 99% تلئل طاطح طحالبح ئتائ لب ) 12امباح طحعط يح ئتائ
استخدام مادة كبريتات النحاس ,Steel and McGhee)ظ د الددديح ي د طحالبحد ط طحاماددخاطحدحا أاتصدداطم طئيمتددبي طحتلددب دألددب يتد
أتص ددددبض تاددددئ تيطمددددا طا طاددددتمب اددددة طحلددددبا امددددباح تاددددئ طحتلددددب اددددة طحي طادددد تددددبم يدددداح ايطاددددبي (1979ل غم 0.5 – 10طحلث لب طحتيططماطاتصالي طحتلب طلاما ح البح اة طحئلميطي طح ت طي طاي ط م لمت تئبمتي
اح ي طلددا لت ددب يددطا طحئلمدديح طحا دددبي طح مامب مددد طحطملمب مدد ح لددبا ئباضددبا طحدد طحصاددب ق /حتددي طيتلددباطم ي دد يدد ح لظ يتا طاتصاطم طحتا طحعبحم لب طئيمتبي طحتلب اأته مؤال طح عتدا (Charles, 1978)طح ماي حمطم ح ب
طحلث دد دد اعدد يتددا طلتاددب طحتددي طااددلبم ئباضددبا طحدد ت طحددا ل ب لدد ح البحدد ضددا طئيمتددبي طحتلددب حددل متااح طحادلم طحتبتلد دحطئيمتبي طحتلب طحخال ااطح طحالبحد طيتلدباطم ي د عبيامد طحلدبا لمدت تتدأثي طحليطئدبي طحلع د
% لدب طحتلدب 55ادأب 7 بي طحتلب لأ طحل طا طحعض م طحلت طتاح ادة طحلدبا يتدا ط مداي تمتة ديب ت بيا طئيمت% لدب طحتلدب طحدلط متلد ا طحد طمد ب طحتلدب 10ادبب 8 طمد ب طحتلدب يتدا أ مداي تمتة طحدلط متلد ا طحد
( (Raman, 1985 طحلاؤ ا يب طحالم طحلتيططل اة طحلبا
188
020406080
100
0 1 2 3 4 5
تركيز كبريتات النحاس ملغم/لتر
رةكو
لع الة
زا ابة
سن
020406080
100
0 1 2 3 4 5
تركيز كبريتات النحاس ملغم/لتر
بالحط ال
لةزا
ابة
سن
كبريتات مع تراكيز للعكورة( تغيرات نسب الازالة 2شكل )
مدة البحثخلال النحاس بثبوت الشبكبريتات مع تراكيز البالطح( تغيرات نسب الازالة 8شكل )
مدة البحثخلال النحاس بثبوت الشب
– 5 ئأيتلبا تي لصت لب طئيمتددبي طحتلدب تتديط ح لدب Jar Testيتا أتديطا تتدبي لصتئيم ئأاتصداطم ت با
ت دا طب طحعطد يح (8 طحيدطا 7لدب )طحيدطا ل غم / حتي حد لظ 30–40ل غم / حتي ئتيي ثبئت لب طحي ئل اطي 1% لدب طحعطد يح طحط مد ئبحتددبحة م ددا 73.4 – 83.3ئلعداا طاطحد متديط ح لدب NTU 14.7–1.12 ئل داطي متديط ح لدب
% معدددا طحادددئ طحددد تددداطصا طئيمتدددبي 100ايا/حتدددي طحددد طحاددد ي أل ئتادددئ أاطحددد 135062أيدددداطا طحالبحددد لدددب لمدت طبتدي أاضدا تادئ أاطحد (Charles,1978) طحتلب طملمب مبم لأ طحليطئبي طحعضد م ح البحد دلط مت دس لدأ
ل غددم / حتددي مت ددس طددلحم لددأ 3 – 4ح عطدد يح طحالبحدد يتددا أاددتصاطم تددي طئيمتدددبي طحتلدددب ئتيطمددا متدديط ح ئددمب (Robert et al.,1980) ئائئ ب تأثيي عمل طحلبلضم ئبحتتبعق لأ امباح تي CuSO4
الاستنتاجـات
حئ تبام م ل اعبحم أي د لدب طحط د يح طائتاط مد لدأ طادتصاطم طحيد ادة تلادمب ط دباح طب أاتصاطم ئيلتطتبي ط -1% أضبا طحد أتد لدباح يمدي ادبل ل تدأثمي ا مدا 99يل م طحتصثمي ااطح طحعط يح طياطا طحالبح ئتائ
للب م ضا طاتصاطل ب ي طئيمتبي طحتلب يتدددا طادددتصاطم ت ددد طحتيططمدددا لدددب pH تبادددم م أعدددا لدددب عدددمم يتدددا طادددتصاطم تيططمدددا ئيلتطتدددبي طحئ pHأب عدددمم -2
طحط ي
References
Charles B. Muchmore, (1978) “Algae Control In Water Supply Reservoirs”, Jour. AWWA, 70,273-
278.
3العدد 7002 ايلول 33المجلد الهندسة مجلة
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Dunst, R. C. (1974). Survey of Lake Rehabilitation Techniques and Experience Tech. Bull. 75. Dept.
Natural Resource, Madison, wis.
Ficek, K. J. (1983). Raw Water Reservoir Treatment with Potassium Permanganate.,74th
Ann. Mtg. III.
Sec. AWWA. Chicago. III. 3 :3:14490.
Hannah, S. A. , Cohen, J. M. and Robeck, G. G. (1967). Control Techniques for coagulation –
Filteration. Jour. AWWA, 59 : 9 : 1149 – 1163.
Janik, J. F. (1980). A Compilation of Common Algal-Control and Management Techniques., Univ.
Nev., Las Vegas. Steel, E. W. and Mc Ghee, T. J. (1979). Water Supply and Sewerage. Fifth edition
McGraw-Hill Book Company, Japan, 285 pp.
Kim Luu (2000) " Study of Coagulation and Settling Processes for Implementation in Nepal "
Submitted To the Department of Civil and Environmental Engineering in Partial Fulfillment of the
Requirements for the Degree Of Master of Engineering in Civil and Environmental Engineering
Kothandaraman, V. & Evans, R.L. (1983) Diagnostic-Feasibility Study of Johnson Sank Trail Lake.,
Contract Rept. 312, State Water Survey, Urbana
Ma, J., Graham, N.J.D., And Li, G.B. (1997) “Effectiveness Of Permanganate Preoxidation In
Enhancing The Coagulation Of Surface Waters-Laboratory Studies” Journal Water SRT-Aqua, 46(1),
1-11.
Mackenthun, K. M. & Kemp, L. E. (1970). Biological Problems Encountered in Water Supplies. Jour.
AWWA , 62:8:520.
Morris , A.W & Riley, J.P. , 1963 . The determination of nitrate in sea water. Analytica chim. Acta ,
vol. 29 , pp. 272 – 297
Prows, B. L. & Mcilhenny, W. F. (1974). Research and Development of a Selective Algicide to
Control Algal Growth. Ofce.Res. & Devel., USEPA, Washington, D.C.
Raman K. Raman, (1985) “Controlling Algae In Water Supply Impoundments” Jour. AWWA, 77:8:Pp.
41-43
Rashash, D., Hoehn, R., Dietrich, A., Grizzard, T., Parker, B. (1996)"Identification and Control of
Odorous Algal Metabolites"J. AWWA Research Foundation and American Water Works Association.
Robert C. Hoehn, Donald B. Barnes, Barbara C. Thompson, Clifford W. Randall, Thomas J. Grizzard,
And Peter T.B. Shaffer, (1980). Algae as Sources of Trihalomethane Precursors. Jour. AWWA,
72:6:Pp.344-350.
Sawyer, C. N. (1968). The Need for Nutrient Control., Jour. WPCF, 40:3:363.
Steel, E. W. and Mc Ghee, T. J. (1979). Water Supply and Sewerage. Fifth edition McGraw-Hill Book
Company, Japan, 285 pp.
Journal of Engineering Volume 13 September2006 Number 3
9461
THE MIGRATION OF LIGHT ORGANIC LIGUIDS IN AN
UNSATURATED-SATURATED ZONE OF THE SOIL
Prof. Dr. Rafa H. Al-Suhaili* Lec. Dr. Ayad A. Faisal
*
University of Baghdad / College of Engineering.
ABSTRACT
A one-dimensional finite difference model for the simultaneous movement of light non-
aqueous phase liquid (LNAPL) and water through unsaturated-saturated zone of the soil in a three
fluid phase system with air assumed constant at atmospheric pressure is developed. The flow
equations described the motion of light non-aqueous phase liquid and water are cast in terms of the
wetting and non-wetting fluid pressure heads respectively. The finite difference equations are solved
fully implicitly using Newton-Raphson iteration scheme. The present numerical results are compared
with results of Kaluarachchi and Parker (1989) and there is a good agreement between them. The
present model can be used to simulate various transport problems in a good manner. Results proved
that the maximum LNAPL saturation occurred below the source of the contaminant during LNAPL
infiltration. During redistribution, the LNAPL saturation had a maximum value at the advancing of
the LNAPL infiltration front.
الخـــلاصـــــة لوصف حركة الماء والسائل العضوي الأخف د يستخدم الفروقات المحددةتم تطوير نموذج عددي ذو بعد واحالدراسة في هذه
منه خلال التربة المشبعة وغير المشبعة مع ثبوت ضغط الهواء عند الضغط الجوي. أن المعادلات التي تصف حركة الماء والسائل ( وباستخدام طريقة fully implicitlyالعضوي وضعت بدلالة عمود الضغط لتلك السوائل. أن معادلات الفروقات المحددة حلت )
(Newton-Raphson تمت عملية اختبار كفاءة النموذج الحالي من خلال مقارنة نتائجه مع نتائج النموذج المقدم من .)أن النموذج الحالي ممكن آن يستخدم لوصف مختلف المسائل بشكل جيد. أن اكبر .Kaluarachchi and Parker (1989)قبل
حدث اسفل مصدر التلوث خلال التسرب في حين تكون هذه القيمة في مقدمة جبهة الملوث خلال اعادة التوزيع.تشبع بالملوث ي
KEYWORDS: Multiphase, Unsaturated, Saturated, Modeling And Contaminant
.
INTRODUCTION
Groundwater contamination due to surface spills or subsurface leakage of Light Non-Aqueous
Phase Liquids (LNAPLs) such as hydrocarbon fuels, organic solvents, and other immiscible organic
liquids is a widespread problem which poses a serious threat to groundwater resources. These
compounds have some acute and long-term toxic effects. As these compounds are migrated through
unsaturated-saturated zone, they will pollute great extents of soil and groundwater. This represents a
major environmental problem. According to the Environmental Protection Agency (EPA) records,
there are 1.8 million underground storage tanks which are in use in the United States. According to
EPA estimates 280000 tanks are leaking, from which more than 20% are discharging their contents
R.H. Al-Suhaili ` The Migration of Light Organic Liguids in an A. A. Faisal Unsaturated-Saturated Zone of the Soil
9461
directly to the groundwater (El-Kadi, 1992). Light Non-Aqueous Phase Liquids (NAPLs) are organic
fluids that are only slightly miscible with water where the “L” stands for “lighter” than water, i.e., less
denser than water.
As the LNAPL (or oil) migrates, the quantity of mobile oil decreases due to the residual oil left
behind. If the amount of oil spilled is small, all of the mobile oil will eventually become exhausted
and the oil will percolate no further. The column of oil is immobile and never reaches the capillary
fringe unless it is displaced by water from a surface source. However, if the quantity of oil spilled per
unit surface area is large, mobile oil will reach the water table. Depending on the nature of the spill, a
mound of the oil will develop and spread laterally. Fig.1 is a pictorial conceptualization of a
subsurface area to which LNAPL is introduced from an oil source, resulting in contamination of the
unsaturated and saturated zones. The LNAPL plume (Fig.1)through the unsaturated zone and it
forms a free-product mound floating on the water table. This mound will spread laterally and move in
the direction of decreasing hydraulic gradient until it reaches residual and can travel no further (Kim
and Corapcioglu, 2003).
Floating LNAPL can partly be removed by using a pumping well. The LNAPL will flow
towards the well facilitated by the water table gradient and can be pumped into a recovery tank. The
movement of oil through unsaturated and saturated zones will accompanied with leaving the residual
droplets (or ganglia) in the pore spaces between the soil particles. This remaining oil may persist for
long periods of time, slowly dissolving into the water and moving in the water phase through
advection and dispersion. In the unsaturated zone, residual oil as well as oil dissolved in the water
phase may also volatilize into the soil gas phase. In the absence of significant pressure and
temperature gradients in the soil gas phase, vapors less dense than air may rise to the ground surface,
while those more dense than air may sink to the capillary fringe, leading to increased contamination of
the saturated zone. Once in the groundwater system, estimates of oil plume migration over time is
necessary in order to design an efficient and effective remediation program. Groundwater modeling
serves as a quick and efficient tool in setting up the appropriate remediation program. In many
regulatory jurisdictions the use of the liquid phase contaminant in an environmental field setting is
prohibited, and numerical modeling is therefore often the only practical alternative in studying the
field-scale behaviour of these compounds (Kim and Corapcioglu, 2003).
A number of multiphase flow models in the contaminant hydrology literature have been
presented. Faust (1985) presented an isothermal two-dimensional finite difference simulator. It
describes the simultaneous flow of water and NAPL under saturated and unsaturated conditions.
Abriola and Pinder (1985) formulated a one-dimensional finite difference model which included
immiscible organic flow, water flow, and equilibrium inter-phase transfer between the immiscible
organic phase, the water phase, and a static gas phase as cited by Sleep and Sykes (1989). Faust et al.
(1989) developed model that might be used to three-dimensional two-phase transient flow system
based on finite difference formulation. The governing equations were cast in terms of non-wetting
fluid (NAPL) pressure and water saturation. Similarly, Parker et al. and Kuppusamy et al. as cited by
(Suk, 2003) developed a two-dimensional multiphase flow simulator involving three immiscible
fluids: namely, air, water and NAPL with the assumption of constant air phase pressure. Kaluarachchi
and Parker (1989) applied a two-dimensional finite element model named MOFA-2D for three phases,
multicomponent, isothermal flow and transport by allowing for interphase mass exchange but
assuming gas phase pressure gradients are negligible.
The present study is aimed to develop a verified multiphase, one-dimensional, finite-difference
numerical simulator which tracks the percentage of LNAPL saturation as well as the lateral and
vertical position of the LNAPL plume in the subsurface at the specified times with different types of
boundary conditions. The present model is formulated in terms of two primary unknowns: wetting
Journal of Engineering Volume 13 September2006 Number 3
9469
phase pressure and non-wetting phase pressure. It is tested on a simple hypothetical problem adopted
by Kaluarachchi and Parker (1989).
Fig.1: Conceptualization representation of LNAPL migration and contamination of the
subsurface.
GOVERNING EQUATIONS
The unsaturated zone is a multiphase system, consisting of at least three phases: a solid phase of
the soil matrix, a gaseous phase and the water phase. Additional phase may also be present such as a
separate phase organic liquid. In the air/oil/water system of the vadose zone, oil is the wetting phase
with respect to air on the surface of the water enveloping the soil grains and the water is the wetting
phase with respect to oil on the soil grain surfaces. The movement of a non-aqueous phase liquid
through the unsaturated-saturated zone may be represented mathematically as a case of two-phase
flow because the air phase equation can be eliminated by the assumption that the air phase remains
constant essentially at atmospheric pressure and consequently the pressure gradients in the air phase
are negligible (Faust et al., 1989). Also, assuming that there is no component partitioning between
liquid phases (Kueper and Frind, 1991).The mass balance equation for each of the fluid phase in
cartesian coordinates can be written as (Bear, 1972): -
(1)
Where f Subscript denoting the phase the equation applies. In the present study, f will refer to water
and oil unless otherwise noted, porosity of the medium, f density of phase f [M][L-3
], fS
saturation of phase f [L3/L
3], fq volumetric flux (or Darcy flux) of phase f [L][T
-1], fQ source or
sink of phase f and t is the time [T]. Darcy’s law is an empirical relationship that describes the
relation between the flux and the individual phase pressure. Since its discovery last century it has been
derived from the momentum balance equations (Kueper and Frind, 1991).
fffff
i
St
Qqx
Impervious Layer
Ground Surface
Spill
Dissolved Plume ( by
Dissolution)
Flow Direction
Vapor Plume( by
Volatilization)
Saturated Zone
Unsaturated Zone
Water
Table
NAPL Core
Capillary
Fringe
R.H. Al-Suhaili ` The Migration of Light Organic Liguids in an A. A. Faisal Unsaturated-Saturated Zone of the Soil
9461
(2)
Where ijk is the intrinsic permeability tensor of the medium [L2], rfk is the relative permeability of
the phase f which is a function to either water (wetting phase) or oil (non-wetting phase). The
relative permeability is a non-linear function of saturation. It ranges in value from 0 when the fluid is
not present, to 1 when the fluid is presented, f dynamic viscosity of fluid f [M][L-1
][T-1
], f fluid
pressure of phase f [M][L-1
] [T-2
], g acceleration due to gravity vector [L][T-2
]. Darcy’s law can be
written equivalently in the form of the pressure head as below:
(3)
With jx
z
is a unit gravitational vector, where z is elevation and t is time. The volumetric flux can be
thought of as the volume of fluid f passing through a unit area of porous medium in a unit time. This
variable is the natural one when making mass fluid balance arguments. By substituting (eq.(3) into
eq.(1)), assuming that the fluid and the porous media are incompressible, ignore the source-sink term,
and according to Kaluarachchi and Parker (1989), the coordinate system is oriented with the
conductivity tensor, or otherwise that off-diagonal components may be disregarded, so that 0 fsij
for i j. the resulting equation can be represented by: -
t
S
x
z
x
h
x
f
w
ff
f
(4)
ONE-DIMENSIONAL NUMERICAL SOLUTION
Eq.4 will be re-written as two one-dimensional equations, one for water phase and the other for
oil phase. Both of which are expressed in terms of the phase pressure head as below:-
t
hC
z
h
z
ww
ww
1
(5)
t
hC
z
h
z
ooro
oo
(6)
Where z is positive upward vertical coordinate [L], rwwsw k and
ro
rowsrooso
kk
; ro is the
ratio of oil to water viscosity, ro is the ratio of oil to water density and C is the specific fluid
capacity, It is defined by w
w
wh
SC
&
o
o
oh
SC
.
However, in the present study, the two-pressure forms of the flow equations are cast in terms of
the two fluid pressure heads. The equations in this form are a direct statement of conservation of mass.
There are two principle difficulties associated with solving the governing equations (eqs.(5) &(6)).
The first is that these equations are first order partial differential equations with respect to time and
second order with space. Further, they are nonlinear differential equations because C and K are
nonlinear functions of capillary pressure heads of two fluids. The second difficulty in solving these
equations lies in the linearization procedure and the method for dealing with the coupling between the
j
f
j
f
f
rfij
fx
zg
x
kkq
jw
f
j
f
fijfx
z
x
hq
Journal of Engineering Volume 13 September2006 Number 3
9461
equations. However, the solution techniques used here consisted of a finite-difference approximation
(implicit method) of the differential equations, the Newton-Raphson with a Taylor series expansion to
treat the nonlinearities, and direct matrix solution (Gauss-Elimination method). These techniques
result in the following equations:-
k
fi
k
fi
k
fi
k
fi
k
fi
k
fi
k
fi rcba
1
1
11
1 (7)
Where:-
k
fia
k
fi
k
fi
rfk
fi
k
fik
fi
k
fih
Rh
hR1
1
1
1
11 21
k
fib
k
jfi
k
jfik
jfi
k
jfi
k
jfik
jfi
k
jfin
jfi
k
jfi
k
jfih
hRRh
ChhC
,
,
,,,1
,
,
,,, 111
k
jfi
k
jfi
rfh
R,
,2
k
fic k
fiR 11
k
firf
k
fi
k
fi
k
fi
k
fi
k
fi
k
fi
k
fi
k
fi
n
fi
k
fi
k
fi
K
fi RhRhRhRhRhhCr 11111 2111111 k
firfR 2
iii zzz
tR
1
1 , 1
11
iii zzz
tR &
iz
tR
2
.
Where k
fir is the residual in node ( i ) at iteration k. At each time step the solution is iterated until the
pressure head converges. During each iteration, the equations are solved to find the pressure heads at
the new iteration level. Thus, the nonlinear coefficients (i.e. C & K) are evaluated using the pressure
heads at the old iteration level thus linearizing the equations. The increment in pressure head at each
iteration level must be added to the tried solution at iteration k, this increment is termed as 1,1 kn , to
produce the solution to the next iteration, 1k and next time step, 1n . Fig.2(a) shows unsaturated-
saturated zone and Fig.2(b) shows the discretized domain in the z-direction with the time.
Fig.2: (a) The common situation to the virtual solution column, (b) image to the solution
domain.
t-coordinate
G.S
W.T
(a)
ΔZ
Δt
n+1
n
i-1
i+1
Z-c
oo
rd
ina
te
i
(b)
G.S is ground surface
W.T is water table U.Z is unsaturated zone S.Z is saturated zone
is known values
U.Z
S.Z
R.H. Al-Suhaili ` The Migration of Light Organic Liguids in an A. A. Faisal Unsaturated-Saturated Zone of the Soil
9466
The coefficients a , b and c in eq. (7) are associated with 1
1
k
i , 1k
i & 1
1
k
i , respectively, and in
general it is formed as:-
kkk rA 1 (8)
Where A is the coefficient matrix for the linearized system. For each node ( i ), there is one linear
equation in three variables 1
1
k
i , 1k
i & 1
1
k
i . The collection of equations for each nodal solution
leads to have a global tri-diagonal coefficient matrix (its bandwidth equal to three) to whole solution
domain. For example, in a ten nodded domain, the soil column is subdivided into ten vertical grids.
Thus, there will be ten equations that form the whole grids. These equations can be written, after
application of the boundary and initial conditions, in matrix notation as:-
kkkr
r
r
r
r
r
r
r
r
r
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
10
9
8
7
6
5
4
3
2
1
110
9
8
7
6
5
4
3
2
1
10
9
10
9
8
9
8
7
8
7
6
7
6
5
6
5
4
5
4
3
4
3
2
3
2
1
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
This system is solved for and then the algorithm enters the next iteration with new values of
fh is evaluated as follows:-
(9)
Where 1k
f is a damping parameter as mentioned by Cooley (1983). During each iteration, nodal
pressure heads need to be updated for the next iteration. With highly nonlinear flow problems, the
updating method introduced by Cooley (1983) which introduces an optimal relaxation scheme, which
accounts for the maximum convergence error for entire mesh, is used in the present study in
conjunction with Newton–Raphson scheme. For clarification, this method is briefly described as
follows:-
Let 1k
fe be the largest in absolute value of the 1k
i values for all i . Then
Step(1):
k
f
k
f
k
f
fe
es
1
for 0k (10)
1fs for 0k
Step(2):
f
f
fs
s
3
3*
for 1fs (11)
f
fs2
1* for 1fs
111 k
fi
k
f
k
fi
k
fi hh
Journal of Engineering Volume 13 September2006 Number 3
9466
Step(3): *1
f
k
f for max
1*
f
k
ff ee (12)
1
max1
k
f
fk
fe
e for max
1*
f
k
ff ee
Where maxfe is the maximum allowable change in the fluid pressure head, fh , during any iteration.
This value is chosen beforehand. The convergence criterion used in the present study for a given
phase f (=o,w) is as follows:-
(11)
Where ε is a small number termed the convergence tolerance. A typical convergence criterion for
pressure head is 0.001 or less.
When the Global finite difference approximation is founded by summing the node equations, the
boundary terms must be accounted. Two types of boundary condition are considered. For Type-1 (
Dirichlet), or fixed head boundary condition, the finite difference equation at boundary node (i) is
replaced by zerok
jfi 1
, or givenhk
jfi 1
, . The other form of boundary condition is Type-2
(Neumann), or specified fluid flux condition. A flux boundary condition is incorporated into the
global approximation at boundary node (i) by using the discretized finite difference form of continuity
equation in conjunction with Darcy’s law.
CONSTITUTIVE RELATIONSHIPS
Eqs.(5) and (6) describe the flow conditions in the subsurface system. There are several
unknowns in these equations. In order to close the system, constitutive relationships that relate the
unknowns must be specified. These relationships can be written in a variety of ways that result in
different variables becoming the dependent variables for the system.
The most common choices for dependent variables are the individual phase pressures and the
phase saturations. The constitutive relationships that will be used in the present study are the
pressure–saturation and relative permeability–saturation relationships. The fluid saturation is a
function of the difference between the pressure of the two fluids in the porous medium, the pressure
difference is called the capillary pressure ,woow , or capillary pressure head woow hhh .
Many different functional forms have been proposed to describe the pressure–saturation and
relative permeability–saturation relationships. They are generally empirical relations. However,
constitutive relationships used in the present study to describe three phase fluid relative permeabilities
and saturations as functions of fluid heads described by (Parker et. al., 1987) which is based on (Van
Genuchten’s model, 1980).The following relationships will be needed to complete the description of a
multiphase flow through the porous media: -
1 aow SSS (14)
owt SSS (15)
1
1
.max
.max
k
fi
k
fi
k
fi
h
R.H. Al-Suhaili ` The Migration of Light Organic Liguids in an A. A. Faisal Unsaturated-Saturated Zone of the Soil
9464
r
rw
wS
SSS
1 (16)
r
rt
tS
SSS
1
(17)
mn
owoww hS
1 .cr
oo hh (18)
mn
aww hS
1 .cr
oo hh (19)
owao
wowcr
o
hh
.
(20)
mn
aoaot hS
1 (21)
2
12
1
11
m
m
wwrw SSk (22)
2
112
1
11
m
m
t
m
m
wwtro SSSSk (23)
Where .cr
oh critical oil pressure head, [L], owh oil-water capillary pressure head wo hh , [L], awh
air-water capillary pressure head wa hh , [L], aoh air-oil capillary pressure head oa hh ,
[L],wS water saturation, aS air saturation, tS total liquid saturation,
wS effective water
saturation,
tS effective total liquid saturation, rwk relative permeability of water, rok relative
permeability of oil, rS residual or irreducible saturation of water phase. Here , n and
nm 11 are
Van Genutchten’s soil parameters, owao & are fluid-dependent scaling coefficients.
NUMERICAL RESULTS IN ONE DIMENSION AND DISCUSSION
A theoretical work presented by Kaluarachchi and Parker (1989) used as a verification to the
present numerical model. They used Galerkin’s finite element method for modeling of one-
dimensional infiltration and redistribution of oil in a uniform soil profile. Also, they used a number of
methods to determine the capacity terms related to oil and water phases. These methods are the chord-
slope scheme and equilibrium scheme.
The problem analyzed here corresponds to a vertical soil column 100 cm long with an oil-free
initial condition in equilibrium with a water table located 75 cm below the top surface. The simulation
was achieved in two stages. The first stage is the infiltration stage, in which oil was allowed to
infiltrate into the column under water equivalent oil pressure head of 3 cm until a total of 5 cm3/cm
2 of
oil had accumulated. The second stage is the redistribution stage, in which the source of oil is cutoff
and oil is allowed to redistribute up to 100 hours. A schematic diagram of the problem is illustrated in
Fig.3. The boundary conditions and system’s parameters, i.e. fluid and soil properties which were
used in this simulation are summarized in Tables (1) and (2) respectively.
Journal of Engineering Volume 13 September2006 Number 3
9461
The initial condition of zero oil saturation was achieved by fixing the initial oil pressure head at
each node to the critical oil pressure head, .cr
oh which is defined in eq. (20). The water saturation
distribution above the water table that is used in this simulation is initially at capillary equilibrium.
This distribution is illustrated in Fig.4. The finite difference mesh consists of 100 grids with a uniform
spacing of 1 cm and the time step varied between 0.00001 to 0.01 hours.
As pointed out by Kaluarachchi and Parker (1989), a jump condition in the water saturation
versus air-water capillary pressure head function, )( aww hS , will occur during the transition from a two-
phase air-water system to a three-phase air-oil-water system. To avoid numerical problems associated
with this jump condition, a phase updating scheme is adopted at the end of each time step to index
whether the node is a two or three phase system (i.e. oil is absent or present). Once a three-phase
condition occurs, reversion to a two-phase condition is not allowed. The criterion for a node to change
from the two to the three phase system is that .cr
oo hh . It is important to note that the part of the
domain remaining as an air-water system and consequently the capacity and relative permeability
terms related to the oil phase oC and rok will become zero and the oil flow equation solution reduce to
the identity 0=0. To avoid this problem, minimum cutoff values of capacity and relative permeability
terms related to the oil phase should be taken as 610oC and 610rok as recommended by
Kaluarachchi and Parker (1989).
The duration of the infiltration stage this is calculated from the present model was
approximately 0.085 hours. While this value was 0.09 hours as calculated by MOFAT-2D model.
There is a good agreement between these values. The total liquid saturation distribution at the end of
the infiltration stage as computed from the present model are compared with those calculated by
MOFAT-2D model by using chord-slope scheme and there is a good agreement between them as
shown in Fig.4. The water saturation distribution at the end of the infiltration stage was identical to
initial condition.
Saturation distributions at the end of 100 hours of redistribution are illustrated in Fig.5 and
Fig.5. These results are compared with those of results MOFAT-2D model by using chord-slope
scheme and equilibrium condition scheme, and, as shown in these figures, there is a good agreement
between these results. Equilibrium distributions were calculated from hydrostatics for an oil volume
of 5 cm subject to the imposed boundary conditions. The equilibrium condition was defined by
Kaluarachchi and Parker (1989) as the fluid distributions at which the total head gradient of both
phases with respect to elevation approaches zero. Kaluarachchi and Parker (1989) pointed out that
convergence of chord-slope results toward the equilibrium results provides a verification check for the
MOFAT-2D numerical model. It is to be noted, however, that whereas the equilibrium distribution
indicates no oil above a depth of 43.0 cm, the MOFAT-2D numerical model, and the present model
predict an average oil saturation of (7-8)% remaining above this depth even though oil velocities at
100 hours are practically zero.
R.H. Al-Suhaili ` The Migration of Light Organic Liguids in an A. A. Faisal Unsaturated-Saturated Zone of the Soil
9461
Fig.3: A schematic diagram for the analyzed problem.
Table1: Boundary conditions which are used for verification.
Table 2: Soil and fluid properties that are used for first verification.
All units are given in centimeters and hours.
Stage
Phase
Boundary
conditions Upper
boundary
lower
boundary
Infiltration
Water
Zero
flux
Constant
head=
25.0 cm
Oil
Constant
head=
3.0 cm
Zero
flux
Redistribution
Water
Zero
flux
Constant
head=
25.0 cm
Oil
Zero
flux
Zero
flux
Parameter
n
rS
sw
ao
ow
ro
ro
Value
3.25
0.05
0.00
50.00
1.80
2.25
0.80
2.00
0.40
Increa
sed d
egree
of
wa
ter sa
tura
tion
ho=3.0 cm H2O
Soil column Soil column
Bottom sealed to both oil and water
Oil
Before oil applying After oil applying
Journal of Engineering Volume 13 September2006 Number 3
9461
0.00 0.20 0.40 0.60 0.80 1.00Percentage of saturation
0
20
40
60
80
100
Dep
th (
cm
)
Sw (@ t=0)
St
Infiltration period, t=0.085 hours
Initial water saturation
MOFAT-2D results
Present results
Fig.4: Distribution of initial water saturation and total liquid saturation at the end of the
infiltration period.
0.00 0.20 0.40 0.60 0.80 1.00Percentage of saturation
0
20
40
60
80
100
Dep
th (
cm
)
Sw
St
Redistribution period, t=100 hours
MOFAT-2D results (Chord-slop scheme)
Present results
Fig.5: Distribution of water and total liquid saturation at the end of the redistribution period.
R.H. Al-Suhaili ` The Migration of Light Organic Liguids in an A. A. Faisal Unsaturated-Saturated Zone of the Soil
9441
0.00 0.20 0.40 0.60 0.80 1.00Percentage of saturation
0
20
40
60
80
100
Dep
th (
cm
)
Redistribution period, t=100 hours
MOFAT-2D results (equilibrium scheme)
Present results
Fig.6: Distribution of water and total liquid saturation at the end of the redistribution period.
The mass balance associated with the above results shows an error in the oil phase during the
infiltration stage is largest at the early time (up to 1% ) and reduce to less than 0.0005% later times
(Fig.7). The present mass balance results can be compared to an analogous lumped finite element
solution in MOFAT-2D using the h-based form of the governing equations. The mass balance error
associated with results of MOFAT-2D is greater than 1% at any time during the infiltration stage. This
is because the finite element approximation may suffer from oscillatory solutions. Such oscillations
are not present in any finite difference solutions. Because the only difference between the two solution
producers as disscused by Celia et al. (1990) is the treatment of the time derivative term, these results
imply that diagonalized time matrices are to be preferred. Thus the unsaturated flow equation is one
that benefits from mass lumping in finite element approximation.
Journal of Engineering Volume 13 September2006 Number 3
9449
0.0 1.0 2.0 3.0 4.0 5.0 6.0Time (min.)
0.4
0.8
1.2
1.6
2.0
Ma
ss b
ala
nc
e r
ati
oPresent results
Fig.7: Mass balance of oil as a function of time, using the h-based form of equations.
MODEL IMPLEMENTATION
A computer program written in DIGITAL VISUAL FORTRAN (Version 5.0) was developed to
implement the model described above. Inherent in any subsurface modeling algorithms are
assumptions and limitations. The major assumptions include:- the pressure in the air phase is constant
and equal to atmospheric pressure, both water and NAPL viscosities and densities are pressure
independent, relative permeability of water is a function of water saturation, relative permeability of
NAPL is a function of air and water saturations, capillary pressure is a function of water saturation, air
saturation is a function of NAPL pressure, Darcy’s equation for multiphase flow is valid, intrinsic
permeability is a function of space and there is no inter-phase mass transfer ( i.e ; the NAPL is truly
immiscible in water).
The major limitations include:- the model can not treat highly pressurized systems in which the
viscosity and density of the three phases are a function of pressure, fractured systems are not treated,
transport of dissolved NAPL is not treated.
CONCLUSIONS
The following conclusions can be deduced :-
R.H. Al-Suhaili ` The Migration of Light Organic Liguids in an A. A. Faisal Unsaturated-Saturated Zone of the Soil
9441
(1) The numerical solution based on the potential form of the governing equations with techniques
consisted of Implicit Finite Difference, Newton-Raphson and Gauss-Elimination schemes showed to
be an efficient procedure in solving one- dimensional water and LNAPL flow through the unsaturated-
saturated zone in three fluid phase's system.
(2) The maximum LNAPL saturation occurred below the source of the contaminant during LNAPL
infiltration. During redistribution, the LNAPL saturation had a maximum value at the advancing of
the LNAPL infiltration front.
3) Mass balance associated with the results presented above show that the finite element
approximation may suffer from oscillatory solutions. Such oscillations are not present in any finite
difference solutions.
REFERENCES
Bear, J., “Dynamics of fluids in porous media”. Elsevier, New York, 1972.
Cary, J.W., J.F. McBride, and C.S. Simmons, “Observations of water and oil infiltration into soil:
Some simulation challenges”. Water Resources Research, 25(1), 73-80, 1989.
Celia, M.A., E.T. Bouloutas, and R.L. Zarba, “A general mass-conservation numerical solution for the
unsaturated flow equation”. Water Resources Research, 26(7), 1483-1496, 1990.
Cooley, R.L., “Some new procedures for numerical solution of variably saturated flow problems”.
Water Resources Research, 19(5), 1271-1285, 1983.
El-Kadi, A.I., “Applicability of sharp-interface models for NAPL transport: 1. infiltration”. Ground
Water, 30(6), 849-856, 1992.
Faust, C.R., “Transport of immiscible fluids, within and below the unsaturated zone : A numerical
model”. Water Resources Research, 21(4), 587-596, 1985.
Faust, C., J. Guswa, and J. Mercer, “Simulation of three-dimensional flow of immiscible fluids within
and below unsaturated zone”. Water Resources Research, 25(12), 2449-2464, 1989.
Kaluarachchi, J.J., and J.C. Parker, “An efficient finite element method for modeling multiphase
flow”. Water Resources Research, 25, 43-54, 1989.
Kim, J., and M.Y. Corapcioglu, “Modeling dissolution and volatilization of LNAPL sources
migrating on the groundwater table”. J. of Contaminant Hydrology, 65, 137-158, 2003.
Kueper, B., and E. Frind, “Two-phase flow in heterogeneous porous media: 1. Model development”.
Water Resources Research, 27(6), 1049-1057, 1991.
Kueper, B., and E. Frind, “Two-phase flow in heterogeneous porous media: 1. Model application”.
Water Resources Research, 27(6), 1059-1070, 1991.
Parker, J. C., and R. J. Lenhard, “A Model for hysteretic constitutive relations governing multiphase
flow: 1. saturation-pressure relations”. Water Resources Research, 23(12), 2187-2196, 1987.
Parker, J. C., R. J. Lenhard, and T. Kuppusamy, “A parametric model for constitutive properties
governing multiphase flow in porous media”. Water Resources Research, 23(4), 618-624, 1987.
Journal of Engineering Volume 13 September2006 Number 3
9441
Marsman, A., “The influence of water percolation on flow of light non-aqueous phase liquid in soil”.
PH.D. Thesis, Wageningen University, Wageningen 2002.
Sleep, B.E., and J.F. Sykes, “Modeling the transport of volatile organics in variable saturated media”.
Water Resources Research, 25(1), 81-92, 1989.
Suk, H., “Development of 2- and 3-D simulator for three phase flow with general initial and
boundary conditions on the fractional flow approach”. PH.D. Thesis, Pennsylvania State University,
2003.
Van Genuchten, M., “A closed-form equation for predicting the hydraulic conductivity of unsaturated
soils’’. Soil Sci. Soc. Am. J.,44,892-898,1980.
SYMBOLS
A Coefficient matrix [M0 L
0 T
0]
fC Specific fluid capacity [M0 L
0 T
0]
maxfe Maximum change in the fluid pressure head [L]
g Acceleration due to gravity vector [L T-2
]
ah Air pressure head [L]
aoh Air-oil capillary pressure head [L]
awh Air-water capillary pressure head [L]
oh Oil pressure head [L]
owh Oil-water capillary pressure head [L]
.cr
oh Critical oil pressure head [L]
wh Water pressure head [L]
i Grid identification in Z coordinates [M0 L
0 T
0]
Hydraulic conductivity [L T-1
]
fs The conductivity when the medium is saturated with fluid f [L T-1
]
rk Relative hydraulic conductivity [M0 L
0 T
0]
k The intrinsic permeability tensor of the medium in eq. (2) [M0 L
0 T
0]
k Iteration index [M0 L
0 T
0]
n Time step identification (if it is superscript) [M0 L
0 T
0]
non The nth
grid identification [M0 L
0 T
0]
nor The number of rows [M0 L
0 T
0]
mn, Van Genuchten’s soil parameters [M0 L
0 T
0]
f Fluid pressure of phase f [M L
-1 T
-2]
fQ Source or sink of phase f [M L
-3 T
-1]
q Volumetric flux (or Darcy’s flux) [L T-1
]
r The residual due to approximation [M0 L
0 T
0]
aS Degree of air saturation %
oS Degree of oil saturation %
rS Degree of residual wetting fluid saturation %
tS Degree of total liquid saturation %
R.H. Al-Suhaili ` The Migration of Light Organic Liguids in an A. A. Faisal Unsaturated-Saturated Zone of the Soil
9446
wS Degree of water saturation %
tS Degree of effective total liquid saturation %
wS Degree of effective water saturation %
t Time coordinate [T]
Van Genuchten’s soil parameter [L-1
]
ij Fluid-dependent scaling coefficient [M0 L
0 T
0]
The difference between the approximation and exact solution [L]
Convergence tolerance [L]
f Dynamic viscosity of fluid f [M L
-1 T
-1]
ro Ratio of oil to water viscosity [M0 L
0 T
0]
f Density of phase f [M L
-3]
ro Ratio of oil to water density [M0 L
0 T
0]
w Density of water at standard temperature and pressure [M L
-3]
Porosity of the medium [L3 L
-3]
Damping parameter [M0 L
0 T
0]
t Time step size [T]
Z Vertical increment in Z-direction [L]
Greek symbols
Journal of Engineering Volume 13 September2006 Number 3
2061
DEVELOPING COMPRESSIBLE TURBULENT FLOW AND
HEAT TRANSFER IN CIRCULAR TUBE WITH UNIFORM
INJECTION OR SUCTION
Asst.Prof.Dr.Ihsan Y.Hussain Ayad M. Salman Adil A. Mohammed
Mech.Engr.Dept. Mechanical Engineer Mechanical Engineer
College of Engineering Energy & Fuel Research Center Baghdad-Iraq
University of Baghdad University of Technology [email protected]
Baghdad-Iraq Baghdad-Iraq
ABSTRACT
In the present work , a numerical study has been made for the developing
compressible turbulent flow and heat transfer in circular tube with uniform injection
or suction. The study included the numerical solution of the continuity, momentum
and energy equations together with the two equations of the (k-ε) turbulence model,
by using the Finite Difference Method (FDM). The air was used as the working fluid,
and the circular passage was composed of tube with diameter (20.0) cm , and the
length was 130 (hydraulic diameter) .The Reynolds number of the flow was
(Re=1.78x106), and the Mach number (M=0.44) the ratio of the transverse velocity at
the wall (vw) to the axial velocity at inlet (Uin), Ω=(vw/Uin), for suction equal(0.001)
and for injection (-0.001).. The wall of the tube was heated with constant wall
temperature (Tw) and in other case with constant heat flux (Qw) as a thermal boundary
condition. The development of both hydrodynamic and thermal boundary layers
occurs simultaneously. The computational algorithm is capable of calculating the
hydrodynamic parameters such as the velocities , friction factor , turbulence structure
which includes the Reynolds stress and the turbulent kinetic energy and eddy
viscosity. Besides, the thermal parameters are also predicted, such as the temperature,
Nusselt number, and the turbulent heat fluxes.The Results showed that the
hydrodynamic and thermal entrance length is increased with the increasing of
Reynolds number. The suction caused a flatten for the velocity profile and thus
decreasing the hydrodynamic entrance length, and caused an increase in the Nusselt
number and decreasing the local coefficient of friction, but injection caused a
steeping of the velocity profile , and thus increasing the entrance hydrodynamic
length and caused a decrease in the Nusselt number and increase the local coefficient
of friction. Turbulent kinetic energy and turbulent heat flux are decreased with
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1603
suction and increased with injection .Predictions have been obtained which are in
good agreement with results obtained by past experimental and theoretical work.
لخلاصةا
يتضمن البحث الحالي دراسة ظررية دددية لياريةان المضةلأرض اطظضةغالأي ميةر تةا التلأةقر مةل اظتةةالا الحةرار ة لا مةل مدةادلتي الحةلا الدةددل لمدةادالا ااسةتمراري قالةا قاللأا ة لأريةة باسةت دا سةحض أق نمل حال حةة أظبقبيمارى
الهةقا ممةا ل باسةت دا ، الحسةابالا الدددية مظفةذ (FDM)رقق المحدد الف باست دا لأرية (k-ε)ظمقذج ااضلأراض، ر ةةة ريظقلةةد ليهةةقا يسةةةاقل ( لأةةر يايةةدرقليمي130قلأقلةةةس يسةةاقل سةة ((20 لأةةر أظبةةقبيدمةةلا يمةةر ةة لا ماةةةرى
1.78x10 بتة مةر شةرلأ دراة حةرار الاةدار ا باسةت دا ، ت تس ين ادار اطسلأقاظ (M=0.44قبر ماخ يساقل (6
مشةةرلأين حةةديين باتاةةايين (vw)مةةذلا اسةةت د مةةرقر الهةةقا دبةةر الاةةدار شةةرلأ ضةةيا حةةرارل ابةةلا باسةةت دا قأ ةةرىقاط ةرى (0.001)قماظةلا الظسةب ( امتصةا اطقلا ضةي حالة ةرقج الهةقا مةن اطظبةقض ,(vw/Uin)=قبظسةب
.تحةدث دميية التلأةقر الهيةدرقديظاميمي قالحةرارل ظيةا . (0.001-)قماظةلا الظسةب (اطظبةقض حةةن إلة قلا الهةقا د ة، ييمةلا ااحتمةااأمماظي الحلا الدددل تتضةمن حسةاض الصةفالا الهيدرقديظاميمية م ةلا مرمبةالا مظحظيةالا السةرد قمدامةلا
ميةة ، مةةذلا تةة حسةةاض الصةةفالا ريظقلةةدا قاللأا ةة الحرميةة المضةةلأرب قالياقاةة الدقا إاهةةادم ةةلا مظحظيةةالا ااضةةلأراضالحراري م لا تقايل دراالا قر ظسةيلا قالفةيا الحةرارل المضةلأرض لمظلأةة الحسةاض . بيظةلا الظتةا ط ايةاد لأةقلا الةد قلا
تسةبض اسةتقا مظحظة السةرد ق صةر لأةقلا اامتصا الهيدرقديظاميمي قالحرارل باياد ر ريظقلدا ، مما بيظلا أن حال إلة المقضدي قالدمة ضة ن الحةةن ية دل ااحتماايمي قاياد ر ظسيلا مما يسبض ظةصان مداملا الد قلا الهيدرقديظام
ااحتمةااتحدض مظحظ السرد قاياد لأقلا الد قلا الهيدرقديظاميمي قاظ فةاا ر ة ظسةيلا مةل ظةصةان ضةي يمة مدامةلا بيظمةا يايةدان ة لا اامتصةا دميية المقضدي . اللأا الحرمية المضةلأرب قالفةيا الحةرارل المضةلأرض يةة ن ة لا
قي مةد ة قمةان التقاضةق بةين الظتةا ط ايةدا لت ميةد الظتةا ط الدددية ضةةد تة مةارظتهةا مةل ظتةا ط البحةقث السةاب الحةن.دميي مق ق ي ال لأقالا الدددي المةترح ضي حسابالا الاريان المضلأرض قاظتةالا الحرار لا المارى اطظبقبي .
KEY WORDS: Flow and Heat Transfer, Developing, Compressible, Turbulent , Injection
and Suction, Circular Tube.
INTRODUCTION The behavior of fluid flow over the surface of a porous material with mass transfer at
the boundary is encountered in a wide range of applications such as, aerodynamic boundary
layer control, wall suction to delay separation and transition from laminar to turbulent flow,
transpiration or sweat cooling of heated surfaces, which is applied to gas turbine blades, ram-
jet intakes, rocket walls, combustion chamber walls exposed to high temperature gases, and so
on. In this cooling method, cooling is forced through a porous wall and injected into the high
Journal of Engineering Volume 13 September2006 Number 3
2061
temperature stream. In this way the wall temperature is reduced by forming a heat insulating
layer between the hot air and the wall. In addition, heat is removed from the wall by the
cooling fluid passing through the interstices. The control of the establishment length in the inlet
region of a channel. It is found that the injection of fluid increases the rate of growth of the
boundary layer .The characteristics of flow with condensation are analogous in many respects
to the flow of fluid over a porous surface with mass transfer at the wall. (Hasan , 1984). For
internal flow through channels with porous walls (suction or injection), there exists many
applications such as in the fields of transpiration cooling, gaseous diffusion, boundary-layer
control and ultrahigh filtration. As an effective boundary layer control method, fluid flow in
channels or pipes with fluid suction or injection through the wall surface was first investigated
by mechanical engineers as early as 1904. Early researchers only focused on fluid flow past a
flat surface suction or injection in a part of the whole surface, it was, assumed that the quantity
of fluid removed from the stream by suction , so small that only fluid particles in the
immediate neighborhood of the wall were suked away, this was equivalent to saying that the
ratio of suction velocity to free stream velocity (Ω) was very small , say (Ω =0.0001 to 0.01) ,
the condition of no slip at the wall is retained with suction present, as well as, the expression
for shearing stress at the wall. (Schilichting , 2000 ).
The present work investigates the effects of injection or suction on the development of
compressible and turbulent flow and heat transfer through circular tube. The governing
continuity , momentum and energy equations are solved numerically by using Finite Difference
Method (FDM). The simultaneous development of both hydrodynamic and thermal boundary
layers will be considered with uniform injection or suction through the wall.
GOVERNING EQUATIONS
Steady state, two dimensional axis-Symmetric , compressible , developing turbulent
flow(both hydrodynamically and thermally), with uniform injection or suction, and negligible
thermal dissipation and body forces and axial diffusion effects will be assumed.
Accordigly the governing, continuity , momentum and energy conservation , kinetic
energy of turbulence (k) and viscous will be as follows :-
Continuity Equation.
01
r
v
rz
u (1)
Momentum equation in radial direction ;
(2)
Where ;
Momentum equation in axial direction ;
rsz
v
zr
v
rerrrr
p
z
vu
r
vv effff
3
2
1
r
k
r
u
tzr
v
rtr
rrrs
3
21 3
2
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1605
(3)
Where; ,
Energy equation
z
T
zr
Tr
rrz
Tu
r
Tv
eff
eff
eff
eff
PrPr
1 (4)
Equation of state (Perfect gas) ;
(5)
Sutherlands law of viscosity
ST
ST
T
T 0
2/3
00
(6)
Where; T0 =273.16 , 0 =1.708x10 kg/m.sec , S=110 .
(k-ε) Model ;
Kinetic energy (k) equation :-
G
z
k
zr
kr
rrz
ku
r
kv
k
t
k
t1 (7)
Viscous dissipation (ε) equation :-
(8)
(9)
Boundary Conditions ;
Entrance condition
u=Uin , v=0 , t=Tin (10)
zsz
u
zr
ur
rrz
p
z
uu
r
uv effeff
1
z
k
z
u
zz
vr
rrzs tt
3
21teff
TP
kCG
kC
zzr
trrrz
ur
v t2
21
1
222
2r
u
z
v
z
u
r
vtG
2
inkin uCk CDkCin hin 5.02
3
Journal of Engineering Volume 13 September2006 Number 3
2060
, (11)
Cε =0.03 , Ck=0.003 , Dh = (12)
Exit Condition
(13 ) tConsz
T
zz
k
z
utan,0
Wall and Center line Conditions
u(R,z)=0 , v(R,z)=vw (14)
Ω = vw / Uin (15)
For the center line:-
r
u
(0,z)=0 , v(0,z)=0 (16)
Temperature boundary condition:-
wTzRT ),( (constant wall temperature) (17)
wQzRr
T
),( ( constant wall heat flux ) (18)
Flow Through Porous Wall
in
w
U
V (20)
m
mwall
D
L
U
V
UD
DLV
in
w
in
w 4
4
2
(21)
*4 Z (22)
NUMERICAL SOLUTION.
The governing equations will be solved numerically by using (Explicit Finite Differences
Method) (Ayad, 2003), the node – point has subscripts ( i , j ) denoting cylindrical
coordinate in ( r , z ) directions. The coordinates for each node are (r=iΔr), (z=jΔz) .
The convection terms of the axial momentum equation will be changed to finite differences
by using back-ward differences. The pressure gradient will be changed to algebraic term by
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1607
using (Upwind Differences). For the (Diffusion Terms) in the right side of the equation, the
(Central Differences ) will be used . The following final form is obtained :-
Suz
ppuauauauaua
jj
jijijijiji
1
1,,11,,1,54321 (23)
where :-
222222
5.0,5.0,5.0,5.0,
,
,5.0,5.0
,
,5.0,5.0,,,,
1zzzzrr
r
rr
r
z
u
r
v jitjitjieffjieff
ji
jieffji
ji
jieffjijijijijia
2
,
,5.0,5.0
2
rr
r
ji
jieffjia
,
2
5.0,5.0,3
z
jijieff ta
,
2
,
,5.0,5.0
4
rr
r
ji
jieffjia
2
5.0,5.0,,,5
zz
u jitjieffjijia
zv
zrr
rv
zrr
tr
vzrr
tr
vzrr
r
Sujijijiji
jiji
jijiji
ji
jiji
jiji
jiji
jiji
jiji t
1,,1,1,
1,1,
,1,11,1
,
,1,1
1,1,
,1,1
1,1,
,1,1
3
1
4444
The mass balance equation for a typical control volume gives ;
mmm wallin ..
(24)
rdrum jiji
R
in ,,
0
2.
(25)
*42.
,,0
Zrdrum jiji
R
(26)
The (+ve) sign in equation (26) is for suction and the (-ve) sign is for injection. The mean
pressure difference ( 1
jjpp ) in equation (23) can be calculated by ( mass conservation
) (Caretto & et.al,1972) as :-
R
ji
R
jiji
jj
rdr
rdrm
pp
0,
0,,
1
2
2.
(27)
The continuity equation is converted to algebraic form by using the (Back Ward
Differences), the following final form is obtained:-
Journal of Engineering Volume 13 September2006 Number 3
2061
)( ,,1,1,
,1,
,,1
,,1
,1,1, jijijiji
jiji
jiji
jiji
jijiji uu
r
r
z
rv
r
rv
(28)
To convert the two equations for (k-ε) model to the numerical form we use the (Backward
Differences) for the convective term and the (Central Differences) for diffusion term , the
result is ;
kSji
kcji
kcji
kcji
kcji
kc
1,,11,,1, 54321 (29)
Where ;
25.0,5.0,
2,
,5.0,5.0
,5.0,5.0
,,,,
1zrr
trr
z
u
r
v
k
jitjit
kji
jiji
jitjijijijiji
c
2,
,5.0,5.0
2rr
r
c
kji
jitji
,
25.0,
3z
c
k
jit
,
2,
,5.0,5.0
,,4
rr
r
r
vc
kji
jitjijiji
25.0,,,
5zz
uc
k
jitjiji
, jijiji
GSk ,,,
Dissipation energy (ε) equation.
Sddddd jijijijiji 1,5,141,3,12,1 (30)
define the following coefficient:-
ji
jijijitjit
ji
jitjijitjijijijiji
k
C
zrr
rr
z
u
r
vd
,
,,2
2
5.0,5.0,
2,
,5.0,5.0,5.0,5.0,,,,1
2,
,5.0,5.0
2rr
rd
ji
jitji
,
25.0,
3z
djit
,
2,
,5.0,5.0,,4
rr
r
r
vd
ji
jitjijiji
25.0,,,
5zz
ud
jitjiji
,
jik
jijiGCS
,
,,1 .
The energy equation can be converted to algebraic form by using the (Backward
Differences) for the diffusion term and (Central Differences) for the convective term , the
following linear equation is obtained:-
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1609
1,5,141,3,12,1 jijijijiji TeTeTeTeTe (31)
Where ;
25.0,5.0,
2,
,5.0,5.0,5.0,5.0,,,,
1z
rpr
rpr
rr
z
u
r
ve
eff
jieffjieff
ji
jieffjijijijijijiji
2,
,5.0,5.0
2r
rpr
r
e
effji
jieffji
,
25.0,
3zp
e
effr
jieff
,
effrji
jieffjijiji
prr
r
r
ve
2,
,5.0,5.0,,4
A numerical calculations algorithm was developed to solve the above equations numerically,
and a computer program was built to implement this algorithm.
RESULTS AND DISCUSSION The results of the developed computational algorithm for turbulent flow of air through a
porous circular tube will be discussed for the following case:-
Pin = 1 bar, Tin = 100 o
C, Tw = 100 o
C, Qw=1000w/m2, Uin= (90-150) m/s, Re= 1.78
x107, M = 0.26 – 0.44, Z
* = 130.
The grid size was taken as, (m=500) in axial direction and (n=20) in radial direction.
Fig.1 shows the development of the boundary layer for flow in solid wall with
( Re=1.78x106) at (Qw=1000w/m
2). Near the wall, the viscous effects are dominant, so the
boundary layer grows in the flow (axial) direction until it reaches apposition ,after that the
boundary shape fixed, this mean that flow is fully developed. It can be concluded that the
hydrodynamic entrance length is equal to (140) pipe diameter approximately. Fig.2 shows the
radial distribution for the axial velocity profile at constant wall temperature and constant heat
flux. Fig.3 show the effect of Reynolds number on the development of velocity profiles at
constant wall temperature and constant heat flux , it is that the velocity increased with the
increasing of Reynolds number, the same result is obtained by (Ayad,2003). Fig.4 shows the
dimensionless axial velocity development for various dimensionless radial positions, with
constant wall temperature and constant heat flux, the same result is obtained by
(Stephenson,1976).
Fig.5 shows the effect of suction and injection on the velocity profiles at constant wall
temperature first and constant heat flux second, the velocity profile is flattened at suction and
became steeper at injection, the same result is obtained by (Hasan,1984). With the presence of
mass transfer through perforations, the velocity profile is altered due to the interaction between
the axial flow and the perforation flow, for the injection case, the injection lifts and expands
the turbulent boundary layer and thus increases the axial velocity beyond the layer while
decreases the velocity within the layer to follow the mass conservation law. As a consequence
the axial velocity near the pipe wall decreases and on the contrary, the suction lowers and
reduces the boundary layer and thus decreases the velocity outside the layer but increases the
velocity inside the layer, and results in an increase of the axial velocity near the pipe wall
analysis, this is consistent with the numerical observations of (Kinney & Sparrow,1970) for
pipe flow with suction through the pipe wall.For suction and injection (+0.001, -0.001) the
Journal of Engineering Volume 13 September2006 Number 3
2026
hydrodynamic length from the entrance to the fully developed velocity profile for the solid
wall at constant wall temperature is equal to (130) diameters , decreases to (120) diameters ,
but for injection increases to (140) diameters, approximately. For Constant heat flux the
hydrodynamic length for the solid wall is equal to (140) diameters , for suction decreased to
(130) diameters , and for injection increased to (150) diameters ,approximately.
Fig.6 shows the development of turbulent kinetic energy ( 2/2 buk )with constant wall
temperature , and constant heat flux , notice that the maximum value of the kinetic energy is in
the region near the wall , and decreases in turbulent kinetic energy value with the increasing of
Reynolds number because the axial velocity profile decreases with the increasing of the
Reynolds number , also the hydrodynamic entry length ,at which the turbulent kinetic energy
is fully developed , increases with the increasing of the Reynolds number. Fig (9) shows the
effect of suction and injection on the turbulent kinetic energy ,turbulent kinetic energy
increased with injection and decreased with suction, the same result is obtained by
(Ayad,2003).
Fig.7 shows the three dimensional development of the turbulent viscosity with. The
turbulent viscosity increases with the increasing of the Reynolds number , so increasing the
length needed for the fully developed profile, the same result is obtained by (Ayad,2003).
Fig.8 show the development of air density for Reynolds number at constant wall
temperature and constant heat flux , notice that for constant wall temperature the density stay
constant near the wall but for constant heat flux decreases because of the increasing of the wall
temperature down stream , the same result is obtained by (Ayad,2003), suction increases the
density and injection decreases it.
Fig.9 shows the steps of developing Reynolds Stress . Find that the Reynolds Stress equal
to zero at maximum velocity near the center line and the maximum value is in the region near
the wall, the same result is obtained by (Ayad,2003). Show that the suction causes to flatten
the velocity profile then decreases the Reynolds stress but the injection which causes to steeper
the velocity profile then decreases the Reynolds stress.
Fig.10 shows the development of the local coefficient of friction , it decreases with the
increasing of the Reynolds number because the boundary layer decreases , and the turbulent
kinetic energy that enter in calculating the shear stress which can be calculated from the wall
function decreases too. Fig.11 shows the effect of suction and injection, the local coefficient
of friction decreases with the suction because that the boundary layer decreases with the
suction case so the turbulent kinetic energy decreases , but in the injection case the local
coefficient of friction increases because the boundary layer increases , the same result is
obtained by (Moshe,1986).
Fig.12 shows development of the isothermal lines, notice that the temperature increases
along side with axial flow direction and decrease with radial direction .Fig.13-a shows the
effect of Reynolds number on the overall heating of the flow. Fig.13-b shows the effect of
Reynolds Number on the wall temperature. The overall heating of the flow and the wall
temperature increase parabolic in the developing region, and decrease with the increasing of
Reynolds number, the same result is obtained by (Ayad,2003).
Fig.15 shows the radial distribution of the turbulent heat flux at many positions , the
turbulent heat flux values is the maximum near the wall and in the fully developed region
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1611
because of the high temperature of the heated wall and there is a decreasing toward the center
line because of the difference in the temperature between the wall and the fluid , the same
result is obtained by (Ayad,2003) .
Fig.16 shows the axial distribution of the Nusselts number at constant wall temperature
and constant heat flux, the maximum value at the entrance region because the thermal
boundary layer thickness equal to zero and the heat transfer coefficient by convection is
maximuim, after that the thermal boundary layer grows and the coefficient of heat transfer
decreases and so the Nusselt number decreases gradually until it reaches constant value , it
needs longer length , also for the same reason the Nusselts number increases with the
increasing of the Reynolds number for constant wall temperature and constant heat flux, the
same result is obtained by (Ayad,2003) . Fig.17 shows the effect of suction and injection on
Nusselts number, suction increases the value of Nusselt Number, the same result is obtained by
(Aggarwal & Hollingsworth,1973) and injection decreases it. Fig,18 shows the radial
temperature development profile at constant wall temperature and constant heat flux . Fig.19
shows the relation between mean Nusselt Number and Rynolds Number, increasing the
Rynolds Number causes significantly increased heat transfer, so increasing Re causing an
increase in Numean , the same result is obtained by (Hasan,1984).
Comparison of the Results .
Fig.20-a shows the comparison of the velocity profile for the present work which is
calculated theoretically with experimental results of (Aggarwal et al,1972), which was done
on a porous tube with internal diameter (D=0.02565 m) and length (L=0.2465)m, for Reynolds
Number Re(101160) and Z* =9.3 with rate of suction Ω =(0.0135 ) . Fig.20-b shows the
comparison of the development of the profile , at (Re=338000) for the present work with the
results of the theoretical work which was done by (Stephenson,1976) , on a tube with (D=0.2
m) with (Re=388000) at
Z* =(0.0 , 0.75, 0.94).
CONCLUSIONS.
The numerical results of the present work show that, the velocity profile, was flattened with
suction and steepened with injection, its value was decreased with the increasing of Reynolds
number and with suction .The hydrodynamic length, from the entrance to the fully developed
region, was increased with the increasing of Reynolds number and with suction, it was
decreased with injection. Turbulent kinetic energy was decreased in the region far from the
wall and by suction, and it was increased by injection. Reynolds stress was vanished far from
the wall; it was increased by injection and decreased by suction. Local coefficient of friction
was decreased with suction and with the increasing of Reynolds number, it was increased with
injection. Turbulent heat flux for constant wall temperature and constant heat flux has a
maximum value near the wall and minimum value in the center line. Decreasing of the wall
temperature at constant heat flux and the bulk temperature increases with injection and
decrease with suction. Nusselts number was increased with the Reynolds number and with
suction, and it was decreased with injection.
Journal of Engineering Volume 13 September2006 Number 3
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REFERENCES
Aggarwal,J.K,(1973) “ Heat Transfer for Turbulent flow with Suction in a Porous Tube ” J. Heat
Mass Transfer , Vol.16.pp.591-609, 1973.
Aggarwal,J.K., Hollingsworth,M.A and Mayhew,Y.R , (1972) “ Experimental Friction Factors
for Turbulent Flow with Suction in a Porous Tube ” J. Heat Mass Transfer. Vol.15.pp.1585-
1602, 1972 .
Ayad Mahmoud Salman , (2003) , “ Turbulent Forced Convection Heat Transfer in the
Developing Flow Through Concentric Annuli ” Msc. thesis , Mechanical Engineering
Department . University of Technology.
Caretto,L.S., Curr,R.M and Spalding,D.B, (1972), “Two Numerical Methods For Three-
Dimensional Boundary Layers,” Compt. Meth. Appl. Mech. Engng., Vol. 1, PP. 39-57.
Hasan Abdul – Aziz Hasan , (1984) , “ Length to Diameter Ratio Effects on Friction and Heat
Transfer of Turbulent Flow in a Porous Tube ” Ph.D. thesis , Mech. Eng. Dept., University
of Bristol.
Kenny,R.B and Sparrow,E.M , (1970) “ Turbulent Flow , Heat Transfer , and Mass transfer in a
Tube with Surface Suction ” ASME J. of Heat Transfer, PP.117-125 , February 1970 .
Moshe Ben-Reuven , (1984) “ The Viscous Wall-Layer Effect in Injected Porous Pipe Flow ”
AIAA Journal , Vol.24.pp.284-294,1984.
Schlichting,H, (2000), “Boundary Layer Theory,”McGraw-Hill, New York.
Stephenson,P.L , (1976) , “ A Theoretical Study of Heat Transfer in Two-Dimensional
Turbulent Flow in a Circular Pipe and Between Parallel and Diverging Plates ” J. Heat Mass
Transfer , Vol.19.pp.413-423, 1976.
NOMENCLATURE
LATIN SYMBOLS
Cf Local coefficient of friction ( = )
Cp Specific heat at constant pressure J / kg .oC
Dh Hydraulic diameter m
G Generation term kg / m. s3
h Heat transfer coefficient w / m .oC
K Von Karman constant
L Length of tube m
m Nodes number in z-direction
25.0/ bw u
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1613
m Mass flow rate kg/s
n Nodes number in r-direction
Nu Nusselts number (= hDh / λ )
P Perimeter
P Pressure N/m2
Pr Prandtle number
Q Heat flux w/m2
r Radial dimension
Re Reynolds number (= UinDh/μ)
R* Dimensionless radial distance( = r/R )
Radial velocity m/sec
T Temperature oC
u Axial velocity m/s
ub Axial Bulk velocity m/s
y Dimensionless distance from the wall
z Axial Cartesian coordinate m
Z*
Dimensionless axial length ( =z/D)
Greek Symbols
Coefficient of relaxation
Ω Velocity ratio (= vw /Uin)
Dissipation rate of turbulent kinetic energy m2/s
3 k
Turbulent kinetic energy m2/s
2
Fluid thermal conductivity W/m.oC
Viscosity kg/m. s
Dimensionless temperature
Kinematic turbulent viscosity m / s
Fluid density kg /m3
, Turbulent Prandtle number
Gas Constant J/kg.K
τw Wall shear stress N /m2
Ψ Fractional mass extraction (= )
v
t
inw mm /
.
Journal of Engineering Volume 13 September2006 Number 3
2021
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1615
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.2 : Developing Axial Velocity for Solid Wall
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.3: Effect of Reynolds Number on Developing Velocity
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.4: Developing Axial Velocity for Solid Wall
R*
R
*
*R *R
inU
u
inU
u
inU
uinU
u
*Z
bu
u
*Z
bu
u
Journal of Engineering Volume 13 September2006 Number 3
2020
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.5: Effect of Suction and Injection on Developing Axial Velocity
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.6: Development of Turbulent Kinetic Energy.
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.7: Development of turbulent viscosity for flow with
inU
u
inU
u
*R*R
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1617
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.8: Developing Air Density for Flow with Re=1.78E+06
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.9: Developing Reynolds Stress for Reynolds Number Re=3.57E+06.
2
2
bu
uv
2
2
bu
uv
*Z*Z
*Z
*R
*R*R
fC fC
Journal of Engineering Volume 13 September2006 Number 3
2021
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.10: Effect of Reynolds Number on the Local Coefficient of Friction
(a).
Con
stan
t
Wal
l
Temperature. (b). Constant Heat Flux.
Fig.11: Effect of Suction and Injection on Local Coefficient of Friction.
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.12: Isothermal Lines at Re=1.78E+06 .
w
fC fC
*Z
*Z
*Z *Z
inTwT
inTb
T
*R *R
*Z
*Z
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1619
(a). Overall Heating. (b). Wall Temperature.
Fig.13: Effect of Reynolds Number on Overall Heating and Wall
Temperature.
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.15: Turbulent Heat Flux Developing Curve .
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.16: Effect of Reynolds Number on Local Nusselt Number .
inTb
u
Tv ''
inTb
u
Tv ''
*R *R
Nu Nu
*Z *Z
Journal of Engineering Volume 13 September2006 Number 3
2016
(a). Constant Wall Temperature. (b). Constant Heat Flux.
Fig.17: Effect of Suction and Injection on Nusselt Number
(a). Constant wall temperature. (b). Constant heat flux.
Fig.18: Radial Temperature Developing Profile for Solid Wall
(a). Constant wall temperature. (b). Constant heat flux.
Fig.19: The Relation Between Mean Nusselt Number and Rynolds Number
Nu Nu*Z *Z
*R *R
meanNu meanNu
Re Re
bu
u
bu
u
I. Y.Hussain developing compressible turbulent flow and heat
A. M. Salman nester in circular tube with uniform injection or
A..A. Mohammed suction
1621
(a) (b)
Fig.20: Comparison Between the Present Work and Past Works
*R*Z
Journal of Engineering Volume 13 September2006 Number 3
2671
AAXXIISSYYMMMMEETTRRIICC FFRREEEE VVIIBBRRAATTIIOONN OOFF TTHHIINN PPRROOLLAATTEE
SSPPHHEERROOIIDDAALL SSHHEELLLLSS
Dr. Ahmed A. Hassan, Dr. Ala M. Hussein, Mohammed J. Obaid
College of Engineering, Babylon University
AABBSSTTRRAACCTT
In this paper a detailed study of the theory of free axisymmetric vibration of thin
isotropic prolate spheroidal shells is presented. The analysis is performed according to
Rayleigh – Ritz method. This method as well as an approximate modeling technique were
attempted to estimate the natural frequencies for the shell. This technique is based on
considering the prolate spheroidal as a continuous system constructed from two spherical
shell elements matched at the continuous boundaries. Through out the obtained results it is
found that this method predicted fairly well the natural frequencies of a prolate spheroidal
shell for all values of eccentricities.
الخلاصة
يتناول هذا البحث الدراسة النظرية للاهتزازات الحرة للقشريات نحيفة الجدران البيضوية الشكل المتطاولة رتز .إن هذه -الخواص في جميع الاتجاهات ، وقد أجري التحليل النظري بطريقة رايلي ةالمتناظرة المحور المتشابه
نها تقريبية ، ولكن بالإمكان الاعتماد عليها لحساب الذبذبة الطبيعية لهذه القشريات .الطريقه بالرغم من كو من خلال النتائج وجد إن تلك الطريقة أعطت نتائج جيدة للترددات الطبيعية للقشرة البيضوية المتناظرة المحور لكل
قيم أللامركزية.
KEYWORDS
Spheroidal Shells, Thin Prolate , Free Vibration, Axisymmetric
A. A. Hassan Axisymmetric free vibration of thin prolate
A. M. Hussein
M..J. Obaid
2671
IINNTTRROODDUUCCTTIIOONN
PPrroollaattee sspphheerrooiidd sshheellllss ccaann bbee oobbttaaiinneedd bbyy rroottaattiinngg aann eelllliippssee aarroouunndd iittss mmaajjoorr aaxxiiss,,
sseeee FFiigg..11.. IItt iiss wwoorrtthhyy ttoo iinnddiiccaattee tthhee iinndduussttrriiaall aapppplliiccaattiioonnss aanndd iimmppoorrttaannccee ooff sshheellll
ssttrruuccttuurreess.. TThhiiss iinntteerreesstt wwaass aapppprreecciiaatteedd ttooddaayy iinn aaeerroossppaaccee,, sseeaa vveehhiicclleess iinndduussttrryy aanndd tthhee
ssttrruuccttuurree ooff rroocckkeett ccaann bbee ccoonnssiiddeerreedd aass aa pprroollaattee sshheellllss.. IInn ssuucchh ssttrruuccttuurreess tthhee rreessoonnaannccee
pprroobblleemm mmaayy ooccccuurr,, tthheerreeffoorree tthhee ssttuuddyy ooff ffrreeee vviibbrraattiioonn bbeeccoommee vveerryy iimmppoorrttaanntt ttoo pprreevveenntt tthhee
rreessoonnaannccee aappppeeaarraannccee..
The study of free vibration of prolate shells take a considerable attempt in the
published literature. several investigators, using a variety of mathematical techniques, have
obtained approximate solutions for the natural frequencies of axisymmetric vibrations of thin
prolate spheroidal shells.
(De Maggio and Silibiger 1961) obtained a solution for the torsional vibrations of thin
prolate spheroidal shell in terms of spheroidal angle functions. (Kanins 1963) was concerned
with the vibration analysis of spheroidal shells, closed at one pole and open at the other, by
means of the linear classical bending theory of shells. Frequency equations are derived in
terms of Legender function with complex indices, and axisymmetric vibration of the natural
frequencies and mode shapes are deduced for all opining angles ranging from a shallow to
closed spherical shell. It was found that for all opening angles the frequency spectrum consist
of two coupled infinite sets of modes that can be labeled as bending (or flexural) and
membrane modes. It was also found that membrane modes are practically independent of
thickness, whereas the bending modes vary with the thickness. The same author concerned
with a theoretical investigation of the free vibration of arbitrary shells of revolution by means
of the classical bending theory of shells.
A method is developed that is applicable to rotationally symmetric shells with
meridional variations (including discontinuities) in Young’s modulus, Poisson’s ratio, radii of
curvature, and thickness. The natural frequencies and the corresponding mode shapes of
axisymmetric free vibration of rotationally symmetric shell can be obtained without any
limitation on the length of the meridian of the shell. The results of free vibration of spherical
and conical shells obtained earlier by means of the bending theory. In addition, parapoloidal
shells and sphere-one shell combination are considered, which have been previously analyzed
Journal of Engineering Volume 13 September2006 Number 3
2671
by means of the inextentional theory of shells, and natural frequencies and mode shapes
predicted by the bending theory are given.
(Numergut and Brand 1965) determined the lower axisymmetric modes of prolate
shell with five values of eccentricity. (DiMaggio and Rand 1966) using membrane shell
theory in which the effects of bending resistance are ignored. Their work was distinguished
by applying their solution to constant thickness membrane shell by means of integrating
numerically the equations of motion. It was found that the frequencies associated with higher
modes are strongly dependent on the eccentricity ratio.
(Zhu 1995) based upon general thin shell theory and basic equations of fluid-mechanics; the
Rayleigh-Ritz’s method for coupled fluid-structure free vibrations is developed for arbitrary fully or
partially filled in viscid, irrigational and compressible or incompressible fluid, by means of the
generalized orthogonality relations of wet modes and the associated Rayleigh quotients.
(Wasmi 1997) used the finite element and modal analysis techniques to investigate the
static and dynamic behavior of oblate spheroidal dishes, prolate and the relevant structures.
Different types of elements were considered in one dimension, two dimensions and three
dimensions.
For framed structures, Euler Bernouilli theory, Tiomshenko theory, integrated Tiomshenko
and improved Tiomshenko theories were applied. While for plates and shells, Kerchief’s,
Zienkiewicz and Mindlin theories were applied. The capability of these trenchancies was
investigated in this work to predict the natural frequencies and mode shapes, as well as the static
analysis of framed structures and spheroidal dishes. It was found that the natural frequencies of
oblate and prolate shells have two types of behavior against increasing the shell thickness and
eccentricity, which are the membrane and bending modes. The membrane modes natural
frequencies tend to increase with increasing the eccentricity of oblate, while the bending mode
natural frequencies decrease with increasing the value of eccentricity.
(Aleksandr Korjanik et al. 2001) investigated the free damped vibrations of sandwich
shells of revolution. As special cases the vibration analysis under consideration of damping of
cylindrical, conical and spherical sandwich shells is performed. A specific sandwich shell finite
element with 54 degrees of freedom is employed. Starting from the energy method the damping
model is developed. Numerical examples for the free vibration analysis with damping based on the
proposed finite element approach are discussed. Results for sandwich shells show a satisfactory
agreement with various references solutions.
(Antoine et, al 2002) investigated the linear and nonlonear vibrations of shallow spherical
shells with free edge experimentally and numerically. Combination resonances due to quadratic
A. A. Hassan Axisymmetric free vibration of thin prolate
A. M. Hussein
M..J. Obaid
2671
nonlinearities are studied, for a harmonic forcing of the shell. Identification of the excited modes is
achieved through symmetric comparisons between spatial results obtained from a finite element
modelling, and spectral information derived from experiments.
This investigation deals with the free vibration characteristics of thin elastic prolate
spheroidal shell. The shell is assumed be of isotropic material. The analysis depends on the
Rayleigh _ Ritz method.
MATHEMATICAL ANALYSIS
Through out the review of literature, it is found that even though the governing equations for
shells of revolution are well spelt out, nevertheless, the governing equations for prolate spheroidal
shells are not available, therefore the approximate energy procedure will be followed.
For a shell undergoing deformation in which the normal to the middle surface of
undeformed shell remains straight and of a constant length under deformation, the shell
displacements can be expressed as, (Burroughs 1978):
iwt
iwt
eUtu
eWtw
)(),(
)(),(
(1)
3/π2θ = 3/πθ =
6/π5θ = 6/πθ =
2/πθ =
2a
e=0.98
e=0.86
e=0.45
Fig. (1): Prolate spheroidal co-ordinates
o
Journal of Engineering Volume 13 September2006 Number 3
2677
2
212
h
R
iin 4/12
1
where, denotes the circular frequency. The stress resultants and couples are related to the
displacement of the reference surface by the same expressions derived in appendix A with the
eccentricity set equal to zero. (Kalnins 1963) show that the actual -dependent coefficients of
the variables can be written as:
)()(3
1
xQBxPAW niinii
i
(2)
i
i
CU )1(3
1
)()( xQBxPA niinii
(3)
)()()1( ''3
1
xQBxPACU niinii
i
i
(4)
)1()1(
3
1
ii
i
CR
hEN
)()( xQBxPA niinii
cot)1( iC )()( xQBxPA niinii (5)
.)1()1(
. 3
1
ii
i
CR
hEN
)()( xQBxPA niinii
cot)1( iC )(')(' xQBxPA niinii (6)
ii
i
b CR
DM )1(1
3
12
)()( xQBxPA niinii
cot)1( iC )()( xQBxPA niinii (7)
ii
i
b CR
DM )1(1
3
12
cot)1( )()( xQBxPA niinii (8)
)1()1(13
12
ii
i
b CR
DQ )()( xQBxPA niinii
(9)
Where,
)1/()1(1
1121222
i
i
iC (10)
(11)
(12)
cosx (13)
The value of i s are the three roots of the cubic equation:
A. A. Hassan Axisymmetric free vibration of thin prolate
A. M. Hussein
M..J. Obaid
2676
)1(12 2v
EhDb
01
1)1(1
1
2)1)(1()1(
)1)(1()1((4)1(4
22222
222223
(14)
Where,
E
R222
(15)
And:
(16)
In the above equations Pn(x), Qn(x) are the Legendre functions of the first and second kinds,
respectively, )(xPn , )(xQn
are the derivatives with respect to ( ) for the Legendre functions of the
first and the second kinds, respectively. Ai's & Bi's are arbitrary constants.
The above solutions can be applied to study the free vibration of an elastic spherical shell
bounded in general by any two concentric openings.
As the shell is taken to be closed at the apex ( =0), and since the Legendre function for the
second kind is singular at this point, then the arbitrary constants (Bi’s ) are set equal to zero. For this
reason all terms involving Qn(x) are omitted.
RAYLEIGH-RITZ METHOD
Rayleigh-Ritz method, which is an extension of the Raleigh’s method, helps to
determine the natural frequencies and mode shapes with general boundary condition in
approximate form.
The Rayleigh-Ritz procedure is essentially statement on the ratio of potential energy to
the kinetic energy. At the natural frequency ( ), and assuming separation of variables, the
shell displacements may be written as give by Eq. (1). Substituting these in the strain energy
expression gives:
dzddRRP
h
h
''''
2/
2/
2
0
2
0
sin2
1
(17)
Journal of Engineering Volume 13 September2006 Number 3
2671
ddRRt
w
t
uK
h
h
sin
2
1222
0
2
0
2/
2/
ddRRwuh
K sin)(2
22
2
0
2
0
2
max
D
N
K
Pr
max
max2
Where,
''
2 ''
)1(
E , ''
2 '
)1(
E (18)
and
zko' zk '', (19)
An expression for the maximum strain energy ( maxP ) may be obtained upon taking eiwt
to be
unity and applying the appropriate expressions for , , '
and '
to given by:
ddRR
WUWU
RR
WUR
WU
RR
W
R
U
WU
RR
WU
RR
R
W
R
U
R
hhEP
''sin
)'sin'cos(''sin
2
)'sin'cos()'sin(
1
'
1
''
.''sin
'cos2
''sin
'cos
''
1
12)1(2
2
2
2
2
2
2
222
2
2
22
0
2
0
2max
(20)
The kinetic energy is:
(21)
After integration with respect to (z) and substituting for the appropriate expression, the
maximum kinetic will take the form:
(22)
Equating the maximum kinetic energy to the maximum potential energy, the natural frequency
can be written as:
(23)
A. A. Hassan Axisymmetric free vibration of thin prolate
A. M. Hussein
M..J. Obaid
2671
)()(3
1
i
i
i waw )()(3
1
i
i
iubu
2
r
cos6
sin212)1(
"'''
3
2
""'''
2
4
2
211
iiiiii
jiiii
o
j
n
i
i
n
i
WWWUWUUURR
vh
WWUUUR
h
v
Ehcc
dRR
WWWUWUUURR
v
WWWUUUR
WWWUUUR
WWWUUURR
h
jiiiiiii
jiiiji
jiiiji
jiiiji
sincossincos2
sincos2sin
cos1
sin21
sin
cos2
12
''
2
2
''''
2
2''''
22
2
Where, N and D represent the equations in numerator and denominator, respectively.
Following the procedure of Rayleigh-Ritz’s method, the radial (or transverse) and tangential
displacements can be written in power series form as:
(24)
Where, the ai's and bi's are coefficients to be determined.
The functions Wi(' ), U( , ) satisfy all the geometry boundary conditions of the system.
Eq.(23) is an exact expression for the frequency according to Rayleigh quotient. In order to
use the procedure of Rayleigh-Ritz’s method, Eq. (24) is substituted into Eq.(20) and (22),
then the results are used in Eq. (23).
Now substituting Eq. (24) into Eqs. (20) and (22), and after some mathematical
manipulations, the following equation will results:
(25)
Where,
(26)
''
2
01 1
sin
dRRWWUUcc jiji
n
i
n
i
ji
(27)
An n-term finite sum leads to the estimation of the first frequencies. Eqs. (26) and (27) gives
the physical properties of the shell from the stiffness and mass distribution point of view.
The stiffness and mass of the shell are given by the following two equations respectively:
Journal of Engineering Volume 13 September2006 Number 3
2661
n
i
n
j
ijji
n
i
n
j
ijji
r
mcc
kcc
1 1
1 12
0//
2
D
cDNcND
c
ii
i
nic
D
D
N
c
N
ii
,...,3,2,1,0
'.
.'sin'cos'sin'cos2
'sin'cos2'sin
'cos1
'sin21
'sin
'cos'''2
12
'cos''''''6
'sin''''''212)1(
2
2
2
2
22
2
3
2
4
22
0
2
dRR
WWWUWUUURR
WWWUUUR
WWWUUUR
WWWUUURR
h
WWWUWUUURR
h
WWWUUUR
hhEk
iiiiiiii
jiiiji
jijiji
jiiiji
iiiiiiii
jiiiiiji
and
(28)
''
2
0
sin dRRWWUUhm jijiij
(29)
Then
(30)
The exact frequency is always smaller than the approximate value. In order to minimize the
approximate value, which given by Eq.(30), it should be differentiated with respect to c i and
equating the resulting expression to zero, that is:
,i=1,2,3,……n (31)
The only way in which this equation can equal is zero if the numerator equals zero, since D is
never equal to zero. The numerator can be written in a more useful form as:
(32) =N / D
A. A. Hassan Axisymmetric free vibration of thin prolate
A. M. Hussein
M..J. Obaid
2662
0
33
2
3332
2
3231
2
31
23
2
2322
2
2221
2
21
13
2
1312
2
1211
2
11
mkmkmk
mkmkmk
mkmkmk
It is as given by equation (23) D
Nr 2 , and n is the number of terms in the approximate
solution. The infinite degrees of freedom system has been replaced by an n degree of freedom
system. Therefore, Eq. (31) can be written in a matrix form as:
02 cMK (33) The stiffness and mass are determined at the edge ( 0 ) of the spherical shell using (28
and 29) respectively. The values equated in above equations are then substituted in the
following determinant:
(34)
The value of 2 which make the determinant equal zero represent the natural frequency of
the shell.
RESULTS AND DISCUSSION
In order to confirm the accuracy of the theoretical results, these results are compared
with the available literature due complexity of obtaining a closed form solution for the free
vibration characteristics of a prolate spheroid shell. From Table (1) it can be noted that the
variation of the natural frequencies of bending modes increase with thickness and with the
mode number. This phenomenon can be elaborated due to the fact that the strain energy
increased with increasing the ratio of thickness for larger eccentricities ratio.
The non-dimensional frequency coefficients for the first three flexural modes which
computed from the present work with (h/a=0.05) are presented in Table (2) along with the
results of (Burroghs and Magrab 1978). From this table it is seen that there is reasonable
agreement between these results, which provide the accuracy of the formulation and results.
Fig.2 shows the non-dimensional natural frequency )/( aE of the first three
modes of vibration as a function of the eccentricity ratio obtained by the Rayleigh- Ritz’s
method using the non-shallow shell theory. It is clearly shown that the tendency of the natural
frequencies towards higher values as the eccentricity ratio increases. This behavior could be
explained by the mode shapes of a closed spherical shell would resemble those of a prolate
spheroid up to certain eccentricity. As the eccentricity increases, the bending stress increased
Journal of Engineering Volume 13 September2006 Number 3
2661
and the potential energy increased. Another reason is that the geometry of the prolate shape
is stiffer than the spherical shape.
Fig.3 gives the first few natural frequencies as a function of the thickness ratio for a
prolate spheroid with (e=0) obtained by RRM. Fig.4 show the first few natural frequencies as
a function of thickness ratio with (e=0.7). All these figures are obtained for )3.0( and they
depend on the bending as well as the membrane modes using the non-shallow shell theory. It
can be noted that the variation of the natural frequency of the bending modes increases with
thickness and with the mode number. This phenomenon can be elaborated due to the fact that
the strain energy increased with increasing the thickness ratio. Also, for larger eccentricity
ratio, the variations are more pronounced than for smaller eccentricities.
Fig.5 shows the effect of eccentricity ratio on the first membrane mode. It is seen that
the natural frequency increased with increasing the eccentricity ratio. The eccentricity ratio
affects the natural frequency hardly at the lower range, while this effect decreased when the
eccentricity ratio beyond 0.8.
The mode shapes of the first three modes of the prolate spheroid shell are shown in
Fig.8, in which both the transverse and tangential displacements are illustrated. This figure
shows that the modes of the transverse displacement occurs at a position in which the
tangential; displacement has maxima and vice versa.
CONCLUTIONS
The main conclutions from the present work can be summarized as;
1- Natural frequencies are seen to have two types of behaviour against increasing the
thickness to major radius ratio. One type, which is associated with the bending modes,
tends to increase with thickness, while the other type, which is associated with
membrane mode, remains unaffected by the thickness variation.
2- Both bending and membrane modes natural frequencies tend to increase with increasing
eccentricity ratio.
3- The natural frequency tends to increase with increasing the ratio of thickness of the
shell.
A. A. Hassan Axisymmetric free vibration of thin prolate
A. M. Hussein
M..J. Obaid
2661
ce
1
cosh
1
vKKv
EhM
vKKv
EhM
vv
EhN
v
EhN
)1(12
)1(12
1
1
2
2
2
2
00
2
00
2
APPENDIX
The principal curvatures of the surface as a function of the angle of inclination ( ) in the following
form.
Where ( ' ) is the angle in the space between the vertical axis and the normal vector, it is given by
22 cos1
sin'cos
e ,
(e) is the eccentricity ratio of the spheroidal shell , which is given by;
The strains, expressed in terms of displacement can be written as:
wu
R '
1
wuR
wuR
'cot1
'sin'cos'sin
1
'
1
'
1 wu
RRk
'
'cos
'sin
1 wu
RRk
If E, are as in nomenclature then, the forces and moments per unit length will be
Substituting the relevant expressions get:-
)cos1(
)cos1(
)1(
22
2/322
2
e
aR
e
eaR
Journal of Engineering Volume 13 September2006 Number 3
2661
)'cot()'
(1
1 2wu
Rw
u
R
hEN
)'
()'cot(1
1 2w
u
Rwu
R
hEN
)'
('sin
'cos)
'(
1
'
1
)1(12 2
3 wu
RR
wu
RR
hEM
))'
(1
('
)'
('sin
'cos
)1(12 2
3 wu
RR
wu
RR
hEM
Table (1): Dimensionless natural frequency coefficients for the axisymmetric free
vibration of a prolate spheroidal shell.
Table (2): Comparison of other estimates of Ω for the flexural modes of a thin prolate spheroidal shell with e=0.7
Mode Number
E=0.3 e=0.7
h/a=0.01 h/a=0.05 h/a=0.01 h/a=0.05
1 0.0 0.0 0.0 0.0
2 0.16 0.16 0.725 0.725
3 0.18 0.19 0.87 0.89
4 0.2 0.23 0.91 0.93
Mode Number Present Work
h/a=0.05
Reference [9]
2 0.725 0.73
3 0.89 0.90
4 0.93 0.95
A. A. Hassan Axisymmetric free vibration of thin prolate
A. M. Hussein
M..J. Obaid
2661
0.01 0.02 0.03 0.04 0.05h/a
0.50
1.00
1.50
2.00
No
n-d
imen
sio
nal fr
eq
uen
cy
Bending theory
Membrane theory
Fig. (4): Effect of the thickness ratio on the natural
frequency of a prolate spheroidal shell (e=0.7)
obtained by RRM
Fig. (3):Effect of the eccentricity ratio on the natural
frequency of a full sphere (e=0) obtained by RRM
Fig.(2): Effect of eccentricity on the first three
bending modes obtained by RRM
Fig.(5): Effect of eccentricity on the first
membrane mode
0.01 0.02 0.03 0.04 0.05h/a
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
No
n-d
imen
sio
nal fr
eq
uen
cy
Bending theory
Membrane theory
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Eccentricity ratio(e)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
No
n-d
ime
nsio
nal
freq
uen
cy
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Eccentricity ratio(e)
1.50
1.75
2.00
2.25
2.50
2.75
3.00
No
n-d
imen
sio
nal fr
eq
uen
cy
Journal of Engineering Volume 13 September2006 Number 3
2667
0 20 40 60 80Conical angle
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
Am
plitu
des/c
oeffecie
nt A
1
W3/A1
U3/A1
(c) Third mode
0 20 40 60 80Conical angle
-1.0
-0.8
-0.5
-0.3
0.0
0.3
0.5
0.8
1.0
Am
plitu
des/c
oeff
ecie
nt
A1
W1/A1
U1/A1
0 20 40 60 80Conical angle
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
Am
plitu
des/c
oeff
ecie
nt
A1
W2/A1
U2/A1
Fig. :(6) Mode shape associated with the first three natural frequency of non-
shallow spheroidal shell (e=0.7) obtained by RRM
(a) First mode (b) Second mode
A. A. Hassan Axisymmetric free vibration of thin prolate
A. M. Hussein
M..J. Obaid
2666
REFERENCES
Aleksandr Korjanik et al, “Free Damped Vibrations of Sandwich Shells of Revolution”, J. of
Sandwich Structures and Materials, Vol. (3), PP. 171-196, 2001.
Antoine, et al, “Vibrations of Shallow Spherical Shells and Gonges”, J. of Sound & Vibration, 2002
Burroughs, C.B. and Magrab, E. B., “Natural Frequencies of Prolate Spheroidal Shells of Constant
Thickness”, J. of Sounds and Vibration, Vol. 57, PP. 571-581, 1978.
Dimaggio, F. L., and Silibiger, A.,”Free Extensional Torsional Vibrations of a Prolate Spheroidal
Shell”,,J.Accoust. Soc.,Amer.,Vol. 33, PP. 56, 1961.
Dimaggio, F.L., and Rand R., "Axisymetrical Vibrations of Prolate Spheroidal Shells", J. Acoust.
Soc. Amer. Vol. 40, PP.179-189, 1966.
Kalnins A. “Free Nonsymmetric Vibrations of Shallow Spherical Shells”, J. of App. Mech., PP.
225-233, 1963.
Kalnins, A. and Wilkinson, F.P., ”Natural Frequencies of Closed
spherical shells”, ,J. App. Mech. Vol. 9,PP.,65, 1965.
Kraus, H.,”Thin Elastic Shells”, John Wiley and Sons, New York 1967
Numergut, P. J., and Brand R.S., “Axisymmetric Vibrations of a Prolate Spheroidal Shell”, J.
Acoust. Soc. Amer., Vol. 38, PP.262-265, 1965.
Shiraishi, N. and Dimaggio, F.L.,”Perturbation Solutions of a Prolate Spheroidal Shells” J. Acoust.
Soc. Amer. Vol. 34, PP. 1725-1731, 1962.
Wilfred, E. Baker “Axisymmetric Modes of Vibration of Thin spherical Shell”, J. Acoust. Soc.
Amer.,Vol.33PP. 1749-1785, 1961.
Wasmi, H. R. Ph.D. Thesis,”Static and Dynamic Numerical and Experimental Investigation of
Oblate and Prolate Spheroidal Shells with & without Framed Structure” Baghdad University, 1997.
Journal of Engineering Volume 13 September2006 Number 3
2661
Zhu, F., "Rayleigh – Ritz Method in Coupled Fluid – Structure Interaction Systems and Its
Applications", J. Sound and Vibration Vol. 186(4), PP. 543-550, 1995.
NOMENCLATURE
iA Arbitrary constants.
a Major semi-axis of a prolate spheroid shell.
iB Arbitrary constants.
b Minor semi-axis of a prolate spheroid shell.
jic , Element of the boundary conditions matrix.
bD Bending stiffness ( )1(12/. 23 vhE ).
E Young's modulus of elasticity (N/2m ).
e Ecentricity ratio(22 /1 ab ).
h Shell thickness(mm).
MM , Moments per unit length (Nm/m).
NN , Membrane forces per unit length (N/m).
Pn(x) Legendre function of the first kind.
Pn’(x) First derivative of the Legendre function of the
first kind.
Pn”(x) Second derivative of the Legendre function of
the first kind.
)(xQn Legendre function of the second kind.
)(' xQn First derivative of the Legendre function of
the second kind.
Q Transverse shearing force per unit length
(N/m).
RR , Principal radii of curvatures of a prolate shell.
t Time (sec).
uu , Tangential displacement (m).
w Transverse or radial displacement (m).
, Strains.
, Inclination angle of a prolate spheroid.
Inclination angle of a spherical shell model.
0 Opening angle of the approximate spherical
shell.
Non-dimensional frequency parameter
(( aE .)/ 2/1 ).
Angle of rotation in the meridian direction
Density (kg/m3
).
Non-dimensional frequency parameter
(( ).)/ 2/1 RE .
Circular frequency (rad/sec).
0 dhE /)/( 2/1 .
, Stress resultants (N/m2
).
Poisson’s ratio.
Journal of Engineering Volume 13 September2006 Number 3
1585
NATURAL CONVECTION FROM SINGLE FINNED TUBE
IMMERRSED IN A TILTED ENCLOSURE
Ass.Prof.Dr.Ikhlase M. Fayed, Hadi R. Roomi
University of Technology / Mech. Eng. Dept.
ABSTRACT
Heat transfer rates of a single horizontal finned tube immersed in water –filled enclosure tilted at
30 degrees are measured .The results serve as a baseline case for a solar water heating system with a
heat exchanger immersed in integral collector storage. Tests were made with both adiabatic and
uniform heat flux boundary conditions. Natural convection flow in enclosure is interpreted from
measured water temperature distributions. Formation of an appropriate temperature difference that
drives natural convection is determined .Correlations for the overall heat transfer coefficient in terms
of the Nusselt and Rayleigh numbers are reduced to the form 0.247
Ra 0.716Nu
7102Ra
5102 For Based on the diameter of the immersed tube .Comparison the present work
results with others gave a good agreement.
KEY WORDS: Solar energy, Natural convection, Heat exchanger
:لخلاصةا ئ ممكيين يلسي قارلييلحبو عرت عي الييقة لحا لحقيي º 03 أابيي م عردي غ عر يي عيلح غقفيي عرلييم ئ قعرليم علحبيي ةين ق يي ق ي نقيس عدييان قال يلحن قةيي ق عي
لح قت ةلحا قاد ن قة ق ي عن جمسع قلجهلحت جي غ ري ح حا ي لبي ت غ جي حي ق قعيلحبع عي قاش لس عبلحغن ح ق ي عر مج ع قا لحق قاش لس أج ت قلافلبقادسض قة ق ي ممكن يلس ج لحن قال لحن قة ق لحة قة غقفي قعرليم عين ت قةلحا قاللحاس قاد ن قة ق ي عن جمسع قلجهلحت علح ةاق قال ح قادل ي إذ لد ض الب
قاللحاس : ق غقفله ئ قادلاق قالج بس لاال لحن قة ق لحة قة ممكن يلسلهلح الاا ةاغ اللت قم ق لي عن فلان قادلاق قالج بس فلان قسلحس غ جلحت قة
Nu=0.716Ra0.247
for 2×105≥Ra≤2×10
7 حص ةلى ا ق جسائقسلت قادلاق ةلى أسلحس ق ق اب م قعر ئ ق ات اللحبو قابحث قةلحلي عع بح ث سلح
INTRODUCTION
One obstacle to the wide spread use of solar water heating systems is either high initial cost .A
potential method of reducing cost is to use components made of plastic rather than metal .One system
concept Fig.1 embodies an expensive bag for collection and storage .An immersed heat exchanger
made of many tubes transfers the stored energy to the potable water circulated through the tubes. The
heat transfer process in the collector and immersed heat exchanger Fig.2 involves the interaction of
negatively buoyant plumes within the tube bundle and a large –scale buoyant flow within the
collector. Natural convection heat transfer characteristics from horizontal tube in an unbounded fluid
have been studied extensively .(Morgan 1975) assembled the widely disparate data and proposed
Nusselt versus Ralyeigh number correlations .Over the range of measured Rayleigh numbers of
interest here, the recommended correlations are
I. M.Fayaed Natural Convection From single Finned Tube
H. R.Roomi Immerrsed in a Tilted Enclosu
1586
7102D
Ra5102For , 0.25 D
Ra 0.480Nu And (1a)
7102
DRa
5102For ,
0.333 D
Ra 0.125Nu (1b)
with a given uncertainty of %5 . (Churchill and Chu 1975) developed a correlation for a wide range
of Raleigh numbers,
1210D
Ra5-10 ,
2
9/169/16(0.559/Pr)1
Ra 387.060.0Nu
(2)
Experiment uncertainty was provided. (Lienhard 1973) obtained a correlation for laminar
convection from a balance of the buoyant and the viscous forces of an arbitrarily shaped immersed
isothermal body,
0.25
Ra 0.52Nu
(3)
Where the length scale is equal to the distance that a fluid particle travels in the boundary layer
on the body .For the horizontal cylinder, 2/ D
Heat transfer in two –dimensional tilted rectangular enclosures with differentially heated surfaces has
been studied for 610Ra310 (Ozoe et al 1975, Sundstrom et al 1996, Canaan et al 1996 Keyhani
et al 1995, Keyhani et al 1996).Similar to the large scale circulating pattern sketched in Fig.2, the
dominant flow is a stable circulating flow that rises along the heating wall, turns and then sinks along
the cold wall. A vertical temperature gradient can develop in the core region of the tilted enclosure
.For 510Ra instabilities in the flow have been found (Hart and J.E. 1971).
Most studies of heat transfer from enclosed bodies, including horizontal tube boundless,
consider cases where the bounding walls drive the convective heat transfer for example,
(Sparrow et al 1983, Warrington et al 1981, Farrington et al 1986, Farrington et al 1986).
Experimental studies of smooth tubes, finned tubes, and coiled tubes immersed in vertical
storage tanks have yielded heat transfer correlations in the form n
Ra CNu (Farrington et al 1986,
Khalillolahi et al 1986).More analogous to the combined ICS/ heat exchanger are studies of transient
natural convection in vertical enclosures with an immersed heat source or sink (Khalillolahi et al
1986).
As a first step toward developing appropriate heat transfer correlations, the heat transfer
coefficient from a single horizontal finned tube in as tilted water-filled enclosure has been measured
in the present work. The temperature field within the enclosure is described using local temperature
measurement .These results provide a base case for a solar water heating system with a heat exchanger
immersed in integral collector storage
APPARATUS
The collector is a rectangular stainless steel enclosure with inside dimensions of 121.9 cm
(width) 94.0 cm (length) 10.2 cm (depth) tilted at ) 30( c
with respect to horizontal Fig.3.
The top side of the enclosure is a removable door to which the heat exchanger and
instrumentation are mounted (3mm diameter) for insertion of the thermocouples are located in the
door and along the bottom face. Additional ports are used to fill and drain the enclosure.
Journal of Engineering Volume 13 September2006 Number 3
1587
A uniform heat flux boundary condition that simulates solar irradiation during charging is
provided by heater attached to the top face of the enclosure.
As indicated in Fig.3, temperatures in the enclosure were measured along the length (z-axis), width (x-
axis) and depth (y-axis) by using copper-constantan thermocouple.
The immersed heat exchanger, shown in Fig.4, is 100.6 cm long and inner and outer diameters
are respectively (19 mm) and (23mm) also the inner and outer diameter for the fins are respectively
(23 mm) and (25 mm), elbows and two 11 cm long vertical brass tubes .The total length of the heat
exchanger is 128.4 cm.
Water temperatures surrounding the tube were measured with four thermocouple probes placed
at
09 increments, 1.1 cm from the outside wall of the tube Fig.5.
A constant temperature flow to the heat exchanger was maintained by the cold water supply,
which includes four electric water heaters to precondition the inlet water and a large water storage
tank. The flow rate was controlled with a gate valve at the outlet of the heat exchanger.
Temperatures, mass flow rate, and power input to the resistance heaters are recorded every 10
minute.
EXPERIMENTAL APPARATUS AND PROCEDURE
Thirteen transient experiments, described in table 1,were conducted over a range of tube flow rates
and levels of simulated insulation .Charge ,discharge and combined charge /discharge modes with
isothermal and stratified initial conditions were investigated .The experiments were begun with the
insulated enclosure filled with hot quiescent water( c 65 ).During the discharge experiments (Nos.1
to 5) water at constant inlet temperature of ) 25( c
flowed through the tube .Flow rates of
0.015,0.030 and 0.050 kg/s were studied .The experiments were continued until water in the
enclosure cooled to about ) 30( c
.The combined charge/discharge experiments (Nos.6 to 12) were
conducted in the same manner except a uniform heat flux was applied to the top face of the enclosure
.These experiments were terminated after 10 hours. The charging experiment (No.13) was conducted
without flow through the tube .It was terminated when fluid at the top of the enclosure
reached ) 90( c
.
DATA ANALYSIS
The data analysis was carried out assuming that the natural convection process is quasi-steady.
At each time step, the overall heat transfer rate of the heat exchanger tube was determined from
measured values of mass flow rate and temperature rise.
)T(TcmQintoup
(4)
Water properties were evaluated at the average of the inlet and outlet bulk temperatures.
The overall natural convection film coefficient was determined from
ncΔT A
Qh
(5)
Where
wncTTΔT
(6)
The value of Tw used in eq. (6) is the average of the wall temperatures at top and bottom of the
tube at the mid –point of its length in the flow direction (thermocouple Nos.26 and 27 in Fig.5)
T was
calculated from thermocouples are (T22, T23, T24 and T25), the Nusselt and Rayliegh numbers at each
time step are determined as
,k
hDNu (7)
I. M.Fayaed Natural Convection From single Finned Tube
H. R.Roomi Immerrsed in a Tilted Enclosu
1588
να
ΔTβD gRa nc
3
(8)
Fluid properties are calculated at the film temperature equal to the average of Tw and
T .The
Nusselt numbers and the Rayleigh numbers are correlated in the form of n
Ra CNu using statistical
software package .Although there are a number of possible choices for the characteristic length, the
data for the single finned tube are found to be well correlated using outer tube average diameter.
RESULTS AND DISCUSION
Measurement of temperature distribution in the enclosure is presented first. The formation of an
appropriate temperature difference that drives free convection is discussed Correlations for the overall
heat transfer coefficient in terms of the Nusselt and Rayleigh numbers are then developed .They
compared to existing correlations for a single tube in an enclosure to determine the effect of the
enclosure for the fin of the tube.
TEMPERATURE DISTRIBUTIONS WITHIN THE ENCLOSURE
Discharge of an initially isothermal enclosure –Results from experiment No.1 to No.3
establish the time –dependence of temperature for an initially isothermal enclosure .Data for
experiment No.1 are presented in Fig.6 to Fig.8. Fig.6 includes data obtained from all thermocouples
in the enclosure as a function of time ;for the duration of the experiment , temperatures are spatially
uniform except near the tube and the lower wall of the enclosure The maximum difference in
water temperature within the enclosure is less than c 3 .Most measurement are within c 1 .
A closer look at the temperatures near the tube for one 30 minute period beginning at t=2 hours
is shown in Fig.7 .The data indicate the existence of slightly hot zone above the tube and a cold plume
sinking from the tube .After two hours, the water 1.1 cm on either side of the tube (thermocouples
Nos.23 and 25)is c 5.1 cooler than that of the region above the tube (thermocouples Nos.21 and 22)
and c 4.2 warmer than the water just below the tube (thermocouples Nos.24 ) .Addititionally ,the
temperatures at the sides of the tube are on average c 2 above those measured in the middle and
bottom portions of the enclosure .The fluid temperatures at the top ,sides and bottom of the tube retain
their relative values over the entire 11.5 hours of experiment No.2.
Readings of the seven thermocouples (Nos.11to7) along the horizontal line in the mid y-z plane
(x=0)are plotted in Fig.8 for the same 30minute period as that of fig .6.The water along the bottom
surface of the enclosure is slightly colder than the bulk of the fluid in the enclosure .The maximum
temperature difference between the reading of the thermocouple located closest to the bottom (No.17)
and the average of the other six thermocouples (Nos.11 to 16)is less than c 2 ,and the average
difference was c 1 .Based on this difference ,we infer that cold water sinking from the tube flows
along the bottom of the enclosure .
Charge with no heat transfer –Heating during charging results in high degree of thermal
stratification in the absence of heat transfer to the tube .The effect of applied heat flux is demonstrated
in experiment No.13,in which the initially isothermal enclosure become thermally stratified when
920W/m2 was applied to the top face of the enclosure .Water temperatures along the mid y-z plane (x
=0)are plotted in Fig.9 for the seven -hour experiment .After seven hours, a temperatures difference of
c 47 existed between positions
z = 4.4 cm (thermocouples No.21) and z =87.6 cm (thermocouple No.7).
Discharge of initially stratified enclosure- In experiments Nos.4 and 5,effect of the initial thermal
stratification on the temperature distributions in the enclosure was investigated .The level of
stratification is characterized by a factor suggested by (Wu et al 1987) equal to the mass weighted
mean square temperature divided by the total mass of fluid in the volume considered:
Journal of Engineering Volume 13 September2006 Number 3
1589
i
i
i
2
avgii
m
)T(Tm
ST (9)
Fig.10 shows the temperature measured during experiment No.4 with the nine thermocouples
located in the mid y-z plane in the enclosure (x =0).At the beginning of this experiment, the top one-
third of the enclosure was filled with c 65
water and the bottom one-third was filled with c 29
water. During the filling process ,the enclosure developed a stratified zone in the vicinity of the tube
.The initial ST is 31.1 K2,based on the topmost one-sixth of the enclosed water volume(0 cm ≤ z ≤15.2
cm).As the experiment proceeded ,temperature measurement indicate that a relatively cool flow sinks
from the tube and flows down the rear surface of the enclosure .After one hour of operation, this wall
plume cools the upper region and middle region the enclosure between z = 0 m and z ~ 0.7 m .It
plunges to the relatively cold lower region between ~ 0.7 m and ~ 0.9 m and heats the water in the
region .Early in the discharge process, the circulating flow is constrained to lie above the colder fluid
near the bottom of enclosure .As the discharge process continues, the region that is colder than the
plume becomes an increasingly smaller fraction of the enclosure volume.
In experiment No.5,the top half of the enclosure was filled with c 65 water and the bottom one-
fourth was filled with c 49 water .After the filling process, the fluid in the region near the tube was
nearly isothermal ;the initial St equals 2.7 K2.Water temperatures are plotted in Fig.11 as a function of
time .Similar to experiment No.4,for which the initial ST is much greater, the wall plume cools the
upper region of the enclosure (0 m to~ 0.6 m from the top )and heats the middle region (~ 0.6 m to~
0.7 m from the top ). Compared with experiment No.4 ,the region cooled by the wall plume is larger
and the not heated any region because of the relatively hot water in the middle and bottom portions of
the enclosure .The momentum of the wall plume is most likely less due to the smaller temperature
difference between the plume and the surrounding water. The circulating flow formed by the wall
plume rapidly grows with time and moves toward the bottom of the enclosure .After an hour of
operation ,the enclosure is isothermal from z ~0.15 to z ~0.8.Once the enclosure becomes isothermal
,it remains so except for a small warmer zone above the tube .
Combined charge/discharge of an initially isothermal enclosure –In the combined charge
discharge experiments, the level of stratification in the enclosure is determined by the strength of the
cold sinking plume relative to that of a warm buoyant wall flow on the front face .In the region near
the tube ,the applied heat flux promotes stratification .The degree of stratification near the tube higher
than that in the previously described discharge experiments with an initially isothermal enclosure .
For experiment No.9 for which the applied heat flux was 960 W/m2,the center portion of the
enclosure , (0.2 m ≤ z ≤0.8 m)is nearly isothermal .On the other hand ,water near the tube is stratified
at the beginning of the experiment and remains so for the duration of the experiment. Fig.12 shows
the water temperature near the tube for 30 minutes after two hours of operation. The temperature of
water at the sides of the tube(thermocouples Nos.23 and 25) is c 3.5
lower than that of the region
above the tube (thermocouples No.22 to 21) and c 7.4
higher than that of the sinking cold plume
(thermocouples No.24) .It is c 1
higher than the temperature measured by the (thermocouples No.20)
.Thus, as opposed to the isothermal discharge experiments, in this case, the temperature measured by
the (thermocouples No.20) does not correctly represent the oncoming temperature that characterizes
natural convection .
Combined charge/discharge with an initially stratified enclosure-Temperatures measured in the
combined charge /discharge experiment with initial thermal stratification (experiment No.12 with an
initial ST=14.1K2)are plotted in Fig.13.The temperature distributions along the length of the enclosure
indicate that the cold plume sinking from the tube descends into the enclosure until it is neutrally
buoyant. Because the plume cannot fully penetrate the lower cold region of the enclosure, the
I. M.Fayaed Natural Convection From single Finned Tube
H. R.Roomi Immerrsed in a Tilted Enclosu
1590
circulating flow gradually extends to the bottom of the enclosure as energy is removed .In the
isothermal region ,the effect of the cold plume on the overall structure of the flow tends to dominate
that of heat input on the front face ,as can be seen by the nearly constant temperatures over the middle
portion of the enclosure for 0.5< t<10 hr .In the portion of the enclosure below the isothermal region
,the enclosure remains stratified ,and the temperature increases due to the addition of heat at the front
face .At the top of enclosure ,near and above the tube ,effects of heat input dominate the flow and
temperature fields.
HEAT TRANSFER CORRELATIONS
Using the statistical software we get the correlation between the Nusselat number and Rayleigh
Number in form 0.247
Ra 0.716Nu ,7
102Ra5
102 For (10)
As shown in Fig.14 .Compare the present work with (Liu et al 2001) then we find the carve of
present work is lower than curve of (Liu et al 2001) with the value [27%] respectively, as shown in
Fig.15.
CONCLUSION
Measurements of water in the enclosure permit interpretation of the flow field. As energy is
transferred to the tube under any level of initial stratification, a cold plume sinks from the tube and
flows to the rear surface of the enclosure 0This plume promotes a circulating flow that promotes
overall mixing in the mid – section of the enclosure. If there is either heat input on the front face or
significant initial stratification near the immersed tube, stratification will persist during the discharge
process.
Measurements of water temperatures near the tube and the tube wall temperature permit
calculation of a Rayleigh number during the cooling process. For 7102Ra5102 , dimensionless
heat transfer coefficients are correlated by 0.247 Ra 0.716Nu .Rayleigh number is based on the
temperature difference between the water temperatures near the tube and the temperature of the tube
wall.
Nomenclature
A out heat transfer area of tube,m2
Cp specific heat of water, J/kg.°c
C Empirical constant used in n
CRaNu
D Tube average diameter ,m
h Natural convection heat transfer coefficient, W/m2.°c
i Enclosure node at constant temperature, used in ST
k Thermal conductivity of fluid,W/m.°c
Length scale used in Ra
m• Mass flow rate in tube ,kg/s
n Empirical exponent in n
CRaNu
Nu Nusselt number,hD/k
Pr Prandtl number, ν/ α
Q• Total energy transferred through the tube bundle ,W
Ra Rayliegh number ,g β D3(Tw-T∞)/ (ν α)
Re Tube Reynolds number
R2 Coefficient of determination
ST Stratification factor ,eq.(9),K2
t Time ,s
Tavg Average or stirred temperature of the enclosure , °c
Ti Water temperature at node I , °c
Tin Water temperature at the inlet of exchanger, °c
Journal of Engineering Volume 13 September2006 Number 3
1591
Tnc Refer to temperature difference that drives natural convection,
°c
Tout Water temperature at the outlet of exchanger, °c
Tw Temperature of outer tube wall, °c
T∞ Storage water temperature that drives natural convection, °c
x,y,z Cartesian coordinates
Greek symbols
α Thermal diffusivity,m2/s
β Coefficient of volumetric thermal expansion,
∆ Refers to temperature difference , °c-1
θ Inclination angle of the enclosure, degrees relative to
horizontal
ν Kinematic viscosity,m2/s
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transfer within horizontal spent-fuel assemblies," Nuclear Technology, 116, No.3, pp. (306-
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convection from a horizontal Cylinder.” University of Penney vania, Journal of Mass and Heat
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Farrington, R.B. and Bingham C.E., "Test and analysis of immersed heat exchangers,"
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Farrington, R.B.," Test Results of immersed coil heat exchangers and liquid storage tanks used in
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Keyhani, M.and Dalton, T., "Natural convection heat transfer in horizontal Rod-bundles
Enclosures," Journal of Heat Transfer, 118, No.3, pp. (598-605) 1996.
Keyhani, M.and Luo, L., "Numerical study of Natural convection heat transfer in horizontal
Rod-bundles, "Nuclear Science and Engineering, 119, No.2, pp. (116-127) 1995.
Khalillolahi,A.,and Sammakia,B., "Unsteady natural convection generated by a heated surface
within an enclosure ," ,Numerical Heat Transfer,9,No.6, pp.(715-730), 1986.
Lienhand, J.H., "On the commonality of equations for natural convection from immersed
Bodies "International Journal of Heat and Mass Transfer, 16, No.11, pp (2121-2123), 1973.
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Application(ICECA'2001),Huazhong University of Science and Technology Press,1.PP(408-413),
Wuhan,China,2001.
Morgan, V. T. ,”The overall convective Heat Transfer from smooth Circular Cylinders
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1592
Sparrow, E.M., and Charm chi, M., “Natural convection Experiments in an Enclosure
between Eccentric or concentric vertical Cylinders of Different Height and Diameter “University
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Table 1.Experimental operating conditions Run
No.
Operating Mode Initial ICS
Temperature
) ( c
Nominal
m
(kg/s)
Re Tin
c
Heat
Flux
(W/m2)
Ra Nu
Δ t
(hr)
1 Discharge 62 0.015 1133~1306 25 0 2×105~9.75×106 15 ~ 33 11.5
2 Discharge 70 0.03 2267~2584 25 0 4.9×105~1.9×107 17.6~42.7 10.8
3 Discharge 75 0.05 3779~4214 25 0 4.1×105~2×107 17.6 ~ 46 10.0
4 Discharge Stratified Top
at 65
Bottom at 29
0.05 3774~3952 25 0 3.4×105~9.7×106 16.7 ~39.5 4..3
5 Discharge Stratified Top
at 65
Bottom at 48
0.05 3774~4076 25 0 4.5×105~8×106 17.6 ~ 40 9.0
6 Charge/Discharge 64 0.03 2420~2581 25 260 4.5×105~8×106 25 ~ 38.6 10.0
7 Charge/Discharge 67 0.03 2420~2581 25 480 1.9×106~1.15×107 27.7 ~ 41.9 10.0
8 Charge/Discharge 75 0.03 2445~2687 25 700 2.12×106~1.23×107 31 ~ 42 10.0
9 Charge/Discharge 62 0.03 2445~2609 25 960 3.38×106~1.2×107 33.5 ~ 41 10.0
10 Charge/Discharge 68 0.15 1276~1370 25 920 4.16×106~9.37×107 30 ~ 38 10.0
11 Charge/Discharge 61 0.05 3912~4119 25 920 4.8×106~1.3×106 32 ~ 37 10.0
12 Charge/Discharge Stratified Top
at 81
Bottom at 44
0.03 2420~2661 25
920 4.47×106~8.19×107 34.5 ~ 49 10.0
13 Charge 27 0 No flow 25 920 - - 7.0
Journal of Engineering Volume 13 September2006 Number 3
1593
City cold water to tube
bundle
Unpressurized bag in glazed
insulated collector
Immersed
tube bundle
Solar heated water
Electric
heating
Mixing
valve
Hot water to
user
Fig (1). Conceptual system for a low cost collector and a load-side heat exchanger
Tube with
cold water
Hotter water
in the bag
Fig (2). Buoyancy driven flow inside the inclined solar collector with an immersed heat exchanger
I. M.Fayaed Natural Convection From single Finned Tube
H. R.Roomi Immerrsed in a Tilted Enclosu
1594
Fig (3). Enclosure and locations of thermocouples Note: All dimensionsin (cm)
To
w su
rface of
(ICS
)
Journal of Engineering Volume 13 September2006 Number 3
1595
Brass tube 11cm
Brass tube 100.6 cm
Nuts for connection with
side door
19mm 23mm
25
mm
Copper elbow
Figure (4). Finned Tube (Heat Exchanger)
Fig (5).The thermocouples around the heat exchanger
I. M.Fayaed Natural Convection From single Finned Tube
H. R.Roomi Immerrsed in a Tilted Enclosu
1596
29
34
39
44
49
54
59
64
0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:00 12:00
TC 1
TC 2
TC 3
TC 4
TC 5
TC 6
TC 7
TC 8
TC 9
TC 10
TC 11
TC 12
TC 13
TC 14
TC 15
TC 16
TC 17
TC 18
TC 19
TC 20
TC 21
41
42
43
44
45
46
47
48
49
50
51
2:00 2:05 2:10 2:15 2:20 2:25 2:30
TC 20
TC 21
TC 22
TC 23
TC 24
TC 25
Time (h:m)
Fig (6). Water temperatures inside the enclosure for experiment No.1
Tem
per
atu
res
(°c)
Time (h:m)
Fig (7). Water temperatures in the top portion of the enclosure over a 30 minute interval during experiment
No. 1
Tem
per
atu
res
(°c)
Journal of Engineering Volume 13 September2006 Number 3
1597
41
42
43
44
45
46
47
48
49
50
2:00 2:05 2:10 2:15 2:20 2:25 2:30
TC 11
TC 12
TC 13
TC 14
TC 15
TC 16
TC 17
20
30
40
50
60
70
80
90
87.477.26756.846.836.82615.24.4
Fig (8). Water temperatures distribution along the horizontal line in the mid y-z plane over a 30 minute interval in
experiment No.1
Tem
per
atu
res
(°c)
Tem
per
atu
res
(°c)
Distance from the top (z), m
Fig (9). Hourly water temperatures distribution in the enclosure during the
charging experiment No.13
I. M.Fayaed Natural Convection From single Finned Tube
H. R.Roomi Immerrsed in a Tilted Enclosu
1598
25
30
35
40
45
50
55
60
65
70
0.8760.7680.6720.570.470.370.2620.1520.06320.05360.044
25
30
35
40
45
50
55
60
65
70
0.8760.7680.6720.570.470.370.2620.1520.06320.05360.044
Distance from the top (z), m
Figure (10). Hourly water temperatures distribution in discharging experiment
No.4
Tem
per
atu
res
(°c)
Tem
per
atu
res
(°c)
Distance from the top (z), m
Fig (11). Hourly water temperatures distribution in discharging experiment No.5
Distance from the top (z), m
Fig (10). Hourly water temperatures distribution in discharging experiment No.4
Journal of Engineering Volume 13 September2006 Number 3
1599
50
52
54
56
58
60
62
64
2:00 2:05 2:10 2:15 2:20 2:25 2:30
TC 20
TC 21
TC 22
TC 23
TC 24
TC 25
40
45
50
55
60
65
70
75
80
85
0.8760.7680.6720.570.470.370.2620.1520.06320.05360.044
Time (h:m)
Figure (21). Water temperatures in the top portion of the enclosure over a 30
minute interval during experiment No. 9
Tem
per
atu
res
(°c)
Tem
per
atu
res
(°c)
Distance from the top (z), m
Fig (13). water temperatures distribution in the mid y-z plane in combined
charge/discharging experiment No.12
Distance from the top (z), m
Figure (13). water temperatures distribution in the mid y-z plane in combined
charge/discharging experiment No.12
Time (h:m)
Fig (21). Water temperatures in the top portion of the enclosure over a 30 minute
interval during experiment No. 9
I. M.Fayaed Natural Convection From single Finned Tube
H. R.Roomi Immerrsed in a Tilted Enclosu
1600
0
10
20
30
40
50
60
70
80
0.00E+00 5.00E+06 1.00E+07 1.50E+07 2.00E+07 2.50E+07
Ra
Nu
0
10
20
30
40
50
60
0.0E+00 5.0E+06 1.0E+07 1.5E+07 2.0E+07 2.5E+07
Ra
Nu
Fig (15).Comparing of the result
Liu, [15]
Present Work
Fig(14). Nusselt number correlation and measured values for
experiments 1-12.
Journal of Engineering Volume 13 September2006 Number 3
4081
INFLUENCE OF DEFECT IN THE CONCRETE PILES USING
NON-DESTRUCTIVE TESTING Prof. Dr. Mosa Al-Mosawe
Department of Civil
Engineering, College of
Engineering, University of
Baghdad.
Prof. Dr. Yousef Al-Shakarchi
Department of Civil
Engineering, College of
Engineering, University of
Baghdad.
Dr. A'amal A-Saidi
Department of Civil
Engineering, College of
Engineering, University of
Baghdad.
ABSTRACT
This paper presents the results of experimental investigation carried out on concrete model
piles to study the behaviour of defective piles. This was achieved by employing non-destructive tests
using ultrasonic waves. It was found that the reduction in pile stiffness factor is found to be about
(26%) when the defect ratio increased from (5%) to (15%). The modulus of elasticity reduction factor
as well as the dynamic modulus of elasticity reduction factor increase with the defect ratio.
الخلاصةستترض هذاتتلبذب التتاذب لرتتاسةذ لم بستتةذب تيرا يتتةذب رتتاذج يتتكذئلتترذلتتتالتذتتت ذ تتاس ذي ستتاليةذتر تت ذي
وم بستتةذرفتت عناذئلتتمذرض تتناذ ملتتتاز ذرتتمذجل تتا ذل تتيذتتت ذيتتمزذب ةلتتمذجيتت ذب رمعتتاذاوبستت ةذب تو تتاكذعتتو ذ%(ذئلتمتاذرت مبمذلستاةذب ت ذتت ذ62)ب فورية ذوقمذايلكذب لرتاسةذات ذتاتمب ذب لافتا ذعتاذتضاتتزذ ستاي ذب يت ذ
%( ذ تتتتتاذج ذئاتتتتتزذب لافتتتتا ذ تضاتتتتتزذب ت ولتتتتةذج تتتتاعةذب تتتترذئاتتتتتزذب لافتتتتا ذاتضاتتتتتزذب ت ولتتتتةذ55%(ذب تتتترذ)5) ب ميلاتي اذي مبمذا يام ذلساةذب
INTRODUCTION
There is a number of factors, which should be considered in the design of bored piles beyond
the routine computation procedures. A review of these factors reveals serious defects, such as, the loss
of continuity along the pile length, and the shaft may contain cracks, voids, inclusion, etc. These
defects may not affect the pile performance in the short term. However, the long-term behavior may
be important, particularly when a pile is subjected to bending stresses (Al-Mosawe and Al-Obaydi,
2002).
M. Al-Mosawe Influence of Defect in the Concrete Piles
Y. Al-Shakarchi using Non-Destructive Testing
A.l A-Saidi
1805
Pile defects can be divided broadly into two categories (Poulos, 1997):
- Geotechnical defects, which arise from either a misassessment of the in-situ conditions during
design or else from construction-related problems.
- Structural defects, which are generally related to construction and which result in the size,
strength and/or stiffness of the pile being less than assumed in design, see Figure (1).
In many cases in the past, it was assumed that the defective pile would not carry any load
and an additional pile or piles have been installed within the group to compensate for the defective
pile. Such a procedure is both costly and time-consuming, and it is therefore of some interest to
examine whether the defective pile can still function satisfactorily. Therefore, a quick non-destructive
method of testing the pile is devised where defects in concrete along the length of the pile could be
estimated to a fair degree of accuracy.
(i) Localized weaker area around some
of the piles
(iii) Soil Depris at base
of pile
(a)
(b)
(i) Necking of the shaft (ii) Low-strength
portion in shaft
(iii) Cracked zone
due to thermal
effects
(iv) Head damage due
to excessive
driving
Necked
area
Damaged
zone
Trapped
bentonite
Weaker
concrete
Cracked
zone
Fig (1) Examples of (a) Typical Geotechnical Defects; (b) Typical Structural Defects (after
Poulos, 1997).
(ii) Trapped Benton
along pile shaft
Journal of Engineering Volume 13 September2006 Number 3
4081
EFFECT OF DEFECTIVE PILE
The presence of defects leads to a reduction in pile axial stiffness; at higher load levels, this
reduction can be severe and gives the appearance of a reduction in axial load capacity. If failure of the
pile occurs because of structural defect, there is a sudden and dramatic increase in settlement. With
geotechnical defects, the apparent loss of load capacity is characterized by a more gradual increase in
settlement with increasing load. The ability of the group to redistribute the pile loads from defective
pile to intact piles results in a less severe reduction in axial stiffness than the case of a single defective
pile. However, the presence of defective piles will generally lead to the development of lateral
deflection and rotation of the group, and induces additional bending moments in the piles, even under
purely axial applied loading (Poulos, 1997).
CONCRETE MIX DESIGN
The ACI 211.1-91 method is used for concrete mixes to obtain the required compressive
strength. The required compressive strength is 30 N/mm2. Mixing proportions were (1:1.85:1.7), and
water - cement ratio (w/c ratio by weight) is 0.5. The water used for mixes was the normal potable
water supplied by the municipality, which was used also for curing concrete samples.
Samples Moulds
Two types of moulds were used for sampling:
- The first type was steel cube moulds (150 mm 150 mm 150 mm). These cubic samples were
used to find the compressive strength of concrete.
- The second type was a plastic cylindrical mould of a diameter (55 mm), with variable lengths
(100 mm, 200 mm, 300 mm, 400 mm, and 500 mm). These samples were used for ultrasonic pulse
velocity tests, and model pile tests.
Ultrasonic Pulse Velocity Test
Ultrasonic pulse velocity test is one of the non-destructive methods to find some of the
physical properties of the concrete; the compressive strength, and the dynamic modulus of elasticity of
concrete. Non-destructive tests reflect the actual properties of concrete, while the destructive tests
(cylinder or cube compression test) carried out on a standard prepared and cured samples, seem to be
too far from the actual conditions.
Many researchers tried to suggest a general limit for the ultrasonic wave velocity. One of
them was Jones and Gatifield (1963) who suggested limits for the ultrasonic wave velocity in
concrete, as given in Table (1):
M. Al-Mosawe Influence of Defect in the Concrete Piles
Y. Al-Shakarchi using Non-Destructive Testing
A.l A-Saidi
1807
Table (1) Velocity a longitudinal ultrasonic pulse for different concrete types (Jones and
Gatifield, 1963).
Concrete Type Pulse velocity (km/sec.)
Very Good More than 4.58
Good 3.66 – 4.57
Moderate 3.05 – 3.66
Poor 2.14 – 3.00
Very Poor Less than 2.14
Concrete Samples Tested:
The dimensions of the cylindrical concrete piles used are listed in Table (2). The ultrasonic
device (Pundit) was used to measure the ultrasonic wave speed through the concrete samples by using
the direct method.
Table (2) Cylindrical Concrete Samples Tested.
Test No. Length of Pile (mm) Type of Pile Type of Defect Defect Ratio Location of Defect
1 100 Sound
2 100 Defected Neck 5% 1/3 length
3 100 Defected Neck 10% 1/3 length
4 100 Defected Neck 15% 1/3 length
5 200 Sound
6 200 Defected Neck 5% 1/3 length
7 200 Defected Neck 10% 1/3 length
8 200 Defected Neck 15% 1/3 length
9 300 Sound
10 300 Defected Neck 5% 1/3 length
11 300 Defected Neck 10% 1/3 length
12 300 Defected Neck 15% 1/3 length
13 300 Defected Neck 5% 1/2 length
14 300 Defected Neck 5% 2/3 length
15 300 Defected External void 5% 1/3 length
16 300 Defected External void 10% 1/3 length
17 300 Defected External void 15% 1/3 length
18 300 Defected External void 5% 1/2 length
19 300 Defected External void 5% 2/3 length
20 300 Defected Internal void 5% 1/3 length
Journal of Engineering Volume 13 September2006 Number 3
4080
21 300 Defected Internal void 5% 1/2 length
22 300 Defected Internal void 5% 1/3 length
23 300 Defected Void
(Internal Void) 1% 1/3 length
24 300 Defected Void
(Internal Void) 1% 1/2 length
25 300 Defected Void
(Internal Void) 1% 2/3 length
26 400 Sound
27 400 Defected Neck 5% 1/3 length
28 400 Defected Neck 10% 1/3 length
29 400 Defected Neck 15% 1/3 length
30 500 Sound
31 500 Defected Neck 5% 1/3 length
32 500 Defected Neck 10% 1/3 length
33 500 Defected Neck 15% 1/3 length
Note:
- Pile diameter (55 mm).
- Defect ratio, is the defect volume compared to the total volume of the pile.
The tests have been performed according to the British standard (BS 1881-Part 203-1086),
and to the American standard (ASTMC597-83-1991).
The ultrasonic test was executed as described above, where the longitudinal wave velocity
was measured by the direct method, and the type of concrete is determined from (Table (1)). The
concrete strength was determined using the following formula (Raouf et al., 1986):
fcu 2.016e0.61V
................................................................................................ (1)
where:
fcu Concrete compressive strength (MPa).
V Wave velocity measured by direct method (km/sec.).
The relation between dynamic modulus of elasticity and the compressive strength of
concrete is descried in (CP110: 1972) by the following equation:
ED 22 + 2.8 cuf ......................................................................................... (2)
where:
ED Dynamic modulus of elasticity for concrete (GPa).
M. Al-Mosawe Influence of Defect in the Concrete Piles
Y. Al-Shakarchi using Non-Destructive Testing
A.l A-Saidi
1809
The expression for the static modulus of elasticity of concrete, Ec, as mentioned in (BS 8110)
is:
Ec 1.25 ED 19 ............................................................................................. (3)
where:
Ec Static modulus of elasticity for concrete (GPa).
Results and Discussion
The results of ultrasonic pulse velocity test are shown in Table (3).
Journal of Engineering Volume 13 September2006 Number 3
4048
Table (3) The Results of Ultrasonic Pulse Velocity Test.
EC103
(MPa) ED10
3
(MPa) fcu
(MPa) Concrete type
(see Table (1) Pulse
Velocity
(km/sec) Location
of Defect Defec
t
Ratio Type of
Defect Type of pile
Length
of pile
(mm)
Test
No.
26.23 36.18 25.66 Good 4.17 Sound 100 1
23.03 33.62 17.23 Moderate 3.52 1/3 length 5% Neck Defected 100 2
18.94 30.35 8.90 Poor 2.44 1/3 length 10% Neck Defected 100 3
16.91 28.73 5.78 Very poor 1.73 1/3 length 15% Neck Defected 100 4
26.21 36.17 25.60 Good 4.17 Sound 200 5
23.27 33.82 17.81 Moderate 3.57 1/3 length 5% Neck Defected 200 6
19.44 30.75 9.77 Poor 2.59 1/3 length 10% Neck Defected 200 7
17.28 29.03 6.30 Very poor 1.87 1/3 length 15% Neck Defected 200 8
25.89 35.92 24.70 Good 4.11 Sound 300 9
23.27 33.82 17.81 Moderate 3.57 1/3 length 5% Neck Defected 300 10
19.37 30.70 9.65 Poor 2.56 1/3 length 10% Neck Defected 300 11
17.32 29.06 6.35 Very poor 1.88 1/3 length 15% Neck Defected 300 12
23.03 33.62 17.23 Moderate 3.52 1/2 length 5% Neck Defected 300 13
22.90 33.52 16.93 Moderate 3.49 2/3 length 5% Neck Defected 300 14
22.95 33.56 17.05 Moderate 3.50 1/3 length 5% External Void Defected 300 15
18.82 30.26 8.70 Poor 2.40 1/3 length 10% External Void Defected 300 16
16.97 28.78 5.86 Very poor 1.75 1/3 length 15% External Void Defected 300 17
M. Al-Mosawe Influence of Defect in the Concrete Piles
Y. Al-Shakarchi using Non-Destructive Testing
A.l A-Saidi
1811
23.44 33.96 18.23 Moderate 3.61 1/2 length 5% External Void Defected 300 18
23.10 33.68 17.4 Moderate 3.53 2/3 length 5% External Void Defected 300 19
Table (3): Continue.
EC103
(MPa) ED10
3
(MPa) fcu
(MPa) Concrete type
(see Table (1) Pulse
Velocity
(km/sec) Location
of Defect Defec
t
Ratio Type of
Defect Type of pile
Length
of pile
(mm)
Test
No.
22.56 33.25 16.14 Moderate 3.41 1/3 length 5% Internal Void Defected 300 20
22.78 33.42 16.64 Moderate 3.46 1/2 length 5% Internal Void Defected 300 21
22.43 33.15 15.85 Moderate 3.38 2/3 length 5% Internal Void Defected 300 22
25.28 35.42 22.98 Good 3.99 1/3 length 1%
Internal
Defect (Gap
in Concrete)
Defected 300 23
25.33 35.47 23.13 Good 4.00 1/2 length 1%
Internal
Defect (Gap
in Concrete)
Defected 300 24
25.07 35.25 22.40 Good 3.95 2/3 length 1%
Internal
Defect (Gap
in Concrete)
Defected 300 25
25.86 35.89 24.60 Good 4.10 Sound 400 26
23.14 33.71 17.50 Moderate 3.54 1/3 length 5% Neck Defected 400 27
18.84 30.27 8.72 Poor 2.40 1/3 length 10% Neck Defected 400 28
16.84 28.67 5.68 Very Poor 1.70 1/3 length 15% Neck Defected 400 29
Journal of Engineering Volume 13 September2006 Number 3
4041
26.09 36.07 25.25 Good 4.14 Sound 500 30
23.40 33.92 18.12 Moderate 3.60 1/3 length 5% Neck Defected 500 31
18.94 30.35 8.90 Poor 2.43 1/3 length 10% Neck Defected 500 32
17.75 29.40 6.99 Very Poor 2.04 1/3 length 15% eck Defected 500 33
M. Al-Mosawe Influence of Defect in the Concrete Piles
Y. Al-Shakarchi using Non-Destructive Testing
A.l A-Saidi
4041
Whenever there is a defect, the pulse velocity decreases, and this means that the mechanical
properties has been affected and transformed to lower strength and lower modulus of elasticity.
The wave velocity for the defect piles was found proportional to defect ratio, and this relation
is not affected by the defect type, Table (3).
The results of ultrasonic tests are shown in Figures (2) to (5). In Figure (2), the relation
between the modulus of elasticity reduction factor, rc (where rc (1 Ecd/Ecs)100, in which Ecd is
the modulus of elasticity for defected pile and Ecs is the modulus of elasticity for sound pile) or the
dynamic modulus of elasticity reduction factor, rd (rd (1 Edd/Eds)100, where Edd is the dynamic
modulus of elasticity for defected pile and Eds is the dynamic modulus of elasticity for sound pile)
and the defect ratio (the defect volume compared to the total volume of the pile).
0 2 4 6 8 10 12 14 16
Defect Ratio
0
10
20
30
40
r
or
rd
c
r
r
c
Figure (7.1) The Relation Between Defect Ratio and Reduction Factor in Static or Dynamic Modulus of Elasticity of Concrete.
d
Fig. (2) The Relation Between Defect Ratio and Reduction Factor in Static or Dynamic
Modulus of Elasticity of Concrete.
It can be noticed that the reduction factors, rc increases to about (45%) while the factor rd
increases to about (20%) when the defect ratio is (15%).
The presence of voids in defected concrete mass causes a reduction in the pulse velocity, V,
of the ultrasonic waves. This is seen in Figure (3) in which the pulse velocity decreases with
increase of defect ratio.
Journal of Engineering Volume 13 September2006 Number 3
4041
0 2 4 6 8 10 12 14 16
Defect Ratio
1.0
2.0
3.0
4.0
5.0
Puls
e V
elo
cit
y v
(k
m/s
ec.)
Figure(7.2 ) The Relation between Defect Ratio and Pulse Velocity.
Fig (3) The Relation Between Defect Ratio and Pulse Velocity.
The pulse velocity decreases to about (50%) when the defect ratio increase from (5%) to
(15%)
Figure (4) shows the relation between the concrete compressive strength reduction factor,
(r) (where r (1 fcud/fcus)100, in which fcud is the concrete compressive strength for defective pile
and fcus is the concrete compressive strength for sound pile) and the defect ratio. It can be noticed
that the factor r increase with the increase of defect ratio. The increase in r is found to be about
(45%) when the defect ratio increases from (5%) to (15%).
0 5 10 15 20
Defect Ratio (%)
0
20
40
60
80
100
Co
ncre
te C
om
pre
ssiv
e S
tren
gth
Red
ucti
on
Fac
tor
(r)
Figure ( ) The Relation Between Defect Ratio and Concrete Compressive reduction Factor (r).
Fig (4) The Relation Between Defect Ratio and Concrete Compressive Reduction Factor (r).
Fig (5) shows a relationship between the defect ratio and pile stiffness reduction factor, Rks
(where Rks (stiffness for defected pile / stiffness) for sound pile)100). The stiffness factor
M. Al-Mosawe Influence of Defect in the Concrete Piles
Y. Al-Shakarchi using Non-Destructive Testing
A.l A-Saidi
4041
decreases with the increase in defect ratio. The reduction in the factor Rks is found to be about
(26%) when the defect ratio increase from (5%) to (15%).
3 6 9 12 15 18
Defect Ratio (%)
8
9
10
11
12
Pil
e S
tiff
nes
s R
educti
on F
acto
r (
R )
ks
Figure ( ) The Relation Between Defect Ratio and Pile Stiffness Reduction Factor ( R ) ks
Fig (5) The Relation Between Defect Ratio and Pile Stiffness
Reduction Factor (Rks).
CONCLUSIONS
The results of this research indicated that when the defect ratio increases from 5 to 15,
then:
a- The pulse velocity decreases to about (50%).
b- The reduction factor rc increases to about (45%), while the factor rd increases to about (20%).
c- The reduction in pile stiffness factor is found to be about (26%).
d- The decrease in concrete strength is found to be about (45%).
References:
-ACI Committee (211.1-91) Standard Practice for Selecting Proportions for Normal, Heavy
weight, and Mass Concrete”, ACI Manual of Concrete Practice, Part 1, 1991.
-Al-Mosawe, M. J. A. and Al-Obaydi, A. H., (2002), "Thermal Analysis in Large Diameter
Bored Piles", 6th
International Conference on Concrete Technology for Developing Countries, 21-
24 October, 2002, Amman, Jordan, pp.763-772.
Journal of Engineering Volume 13 September2006 Number 3
4041
-ASTM C597-83, (1991), "Test for Pulse Velocity Through Concrete", American Society for
Testing and Materials, 1991.
-BS 1881: Part 203: 1986, “Recommendations for Measurement of Velocity of Ultrasonic Pulses in
Concrete”, British Standards Institution, London.
-CP 110, 1972, "The Structural Use of Concrete", Code of Practice, Part 1, "Design Materials and
Workmanship".
-Jones, R. and Gatfield, Exl., (1963), “Testing Concrete by an Ultrasonic Pulse Technique”, London
Her Majesty’s Stationary Office, 1963, pp.15-16.
-Poulos, H. G., (1997), "Behaviour of Pile Groups Containing Defective Piles", Proceeding of 14th
International Conference of Soil Mechanics and Foundation Engineering", Hamburg, Vol.2, pp.670.
ئيتةذب ي ستالةذا ستريمبمذوذيم ذيولس،ذوا يسةذتلتمذفا ح،ذ"ب سي ذئلترذل ذ ي ذب ضاامي ذ ؤوفذتلتم،ذعذ-
ذ 5892اسعذلمو ذب رضليمذب تسرت ،ذقب لالااكذعو ذب فورية"،ذوذ
Journal of Engineering Volume 13 September 2007 Number 3
1780
SACRIFICIAL ANODE CATHODIC PROTECTION OF LOW
CARBON STEEL IN SEA WATER
Dr. Aprael S. Yaro Dr. Khalid W. Hameed
University of Baghdad / College of Engineering
ABSTRACT
Cathodic protection is a corrosion-prevention technique which uses the electrochemical
properties of metals to insure that the structure to be protected becomes the cathode of an
electrolytic cell.
Laboratory evaluation was conducted on zinc electrode as anode material that used for sacrificial
anode cathodic protection (SACP)of carbon steel.
Rate of zinc consumption during cathodic protection of carbon steel pipeline carrying seawater
(4 % w/v NaCl solution) were measured by the loss in weight technique. Variables studied were
seawater temperature (0-45o C), flow rate (5-900 lit/h), pH (2-12) and duration time (1-4 h). It was
found that the rate of zinc consumption increases with increasing seawater temperature, flow rate
and duration time and decreases with pH increase. Under the mentioned operating conditions, the
rate of zinc consumption during cathodic protection of steel ranged from 5.65×10-3
to 98.9×10-3
g/cm2.day.
For the system under investigation, the cell responsible for cathodic protection is Zn/NaCl/Fe.
INTRODUCTION
Corrosion is an electrochemical process in which a current leaves a structure at the anode site,
passes through an electrolyte and reenters the structure at the cathode site. Current flows because of
a potential difference between the anode and cathode that is the anode potential is more negative
than the cathode potential, and the difference is the driving force for the corrosion current. The total
A. S. Yaro sacrificial anode cathodic protection
K. W. Hameed of low carbon steel in sea water
1781
system-anode, cathode, electrolyte and metallic connection between anode and cathode is termed a
corrosion cell (Shrier(1), 1976 and Halford, 1985).
Corrosion control is a process in which humans are very much in control of materials and
environments can regulate the rate of corrosion, keeping it within acceptable or at least predictable
limits for life of the structure (Trethewey and Chamberlain, 1996).
There are many methods for corrosion control as illustrated some of them (Bosich, 1970 and
Schweitzer, 1987),( cathodic protection, anodic protection ,protective coating such as paint
,corrosion-resistant metals and alloys ,addition of inhibitors , very pure metals, ….etc).
The selecting method depends on many factors such as cost,efficiency, availability, contamination
of environment with corroding metal, …etc.
Cathodic protection is unique amongst all the methods of corrosion control in that if required it is
able to stop corrosion completely, but it remains within the choice of the operator to accept a lesser,
but quantifiable, level of protection. Manifestly, it is an important and versatile technique. In
principle, cathodic protection can be applied to all the so-called engineering metals.
In practice, it is most commonly used to protect ferrous materials and predominantly carbon steel. It
is possible to apply cathodic protection in most aqueous corrosive environments, although its use is
largely restricted to natural near-neutral environments (soils, sands and waters, each with air
access). Thus, although the general principles outlined here apply to virtually all metals in aqueous
environments, it is appropriate that the emphasis, and the illustrations, relate to steel in aerated
natural environments (Shrier, (2), 1994 and Almardy, 1999).
Cathodic protection involves the application of a direct current (DC) from an anode through the
electrolyte to the surface to be protected. This is often through of as "overcoming" the corrosion
currents that exist on the structure. There is no flow of electrical currents (electrons) through the
electrolyte but flow of ionic current. Cathodic protection eliminates the potential differences
between the anodes and cathodes on the corroding surface. A potential difference is then created
between the cathodic protection anode and the structure such that the cathodic protection anode is
of a more negative potential than any point on the structure surface. Thus, the structure becomes the
cathode of a new corrosion cell. The cathodic protection anode is allowed to corrode; the structure,
being the cathode, does not corrode (Brown and Root, 2003 and Jezmar, 2003).
There are two proved methods of applying cathodic protection: sacrificial anode (galvanic) and
impressed current. Each method depends upon a number of economic and technical considerations
and as certain advantages. For every structure, there is a special cathodic protection system
dependent on the environment of the structure (Scully, 1975).
Journal of Engineering Volume 13 September 2007 Number 3
1782
Current distribution in cathodic protection system is dependent on several factors, the most
important of which are driving potential, anode and cathode geometry, spacing between anode and
cathode and the conductivity of the aqueous environment which is favorable toward good
distribution of current (Shrier (2), 1994)
Structures commonly protected are the exterior surfaces of pipelines, ships' hulls, jetties, foundation
piling, steel sheet-piling and offshore platforms. Cathodic protection is also used on the interior
surfaces of water-storage tanks and water-circulating systems. However, since an external anode
will seldom spread the protection for a distance of more than two or three pipe-diameters, the
method is not suitable for the protection of small-bore pipework. Cathodic protection has also been
applied to steel embedded in concrete, to copper-based alloys in water systems and exceptionally to
lead-sheathed cables and to aluminum alloys, where cathodic potentials have to be very carefully
controlled (Halford, 1985 and Davies, 2001).
The present work considered the sacrificial anode cathodic protection. The effect of temperature
(0-45o C), flow rate (5-900 lit/h), pH (2-12) and time (1-4 h) on the rate of zinc consumption during
cathodic protection of steel tube exposed to seawater (4 %w/v NaCl in distilled water). The rate of
zinc consumption was determined by the loss in weight technique.
EXPERIMENTAL WORK
The apparatus shown in figure (1) was used to study the variables, temperature (0-45o C),
flow rate (5-900 lit/h), pH (2-12) and time (1-4 h) in the sacrificial anode cathodic protection
system.
The container vessel was filled with seawater (4% (w/v) NaCl), then adjusting the pH and
temperature to the desired values. Before each run, the zinc strip was weighted and fixed at the inlet
of the steel tube by rubber stopper and was electrically connected by an insulated copper wire to the
steel tube outlet as shown in figure (1).The zinc strip is extending along the steel tube to ensure
uniform current and potential distribution along the tube wall.
The seawater was pumped from the vessel by the pump through the rotameter to measure the
desired flow rate, then the seawater inter from the below the steel tube and out from upper the steel
tube to return to the vessel again (i.e. the seawater is circulated between the vessel and steel tube for
desired time).After each run the zinc strip was rinsed in distilled water and brush to remove the
corrosion products, dried with clean tissue then immersed in the benzene and acetone, dried again,
and then re-weighted to determine the weight loss. The steel tube is also rinsed and dried by the
same way above in order to re-use another time. After each run the vessel is emptied from the
solution and washed with distilled water, then filled with a new prepared solution for new run.
A. S. Yaro sacrificial anode cathodic protection
K. W. Hameed of low carbon steel in sea water
1783
Fig. 1, Schematic diagram of apparatus used in sacrificial anode test system
RESULTS AND DISCUSSION
Time effect
Figures 2,3 and 4 show the rate of zinc consumption (dissolution), as an instead of corrosion
rate of steel, with time at different temperatures, different flow rates and different pH, respectively.
Where the rate of zinc dissolution increases with increasing time and this is normal case. But this
increasing is not equally with time, where the dissolution rate in the first hour is more than second
hour and so on. The reasons of that are attributed to continuous growth of the corrosion products
layer with time, which affects the transport of oxygen to the metal surface and the activity of the
surface and hence the corrosion rate. Also, the cathodic reactions will result an increase in pH with
time either by the removal of hydrogen ions (
222 HeH ) or by the generation of hydroxyl
ions ( OHHeOH 222 22 and OHeOHO 442 22 ) both reasons are reduced the
corrosion rate of steel and hence the dissolution rate of zinc.
Temperature effect
Figures 2, 5 and 6 show the effect of temperature on the rate of zinc dissolution with time
with different flow rates and with different pH’s, respectively. The increase in the rate of zinc
dissolution with increasing seawater temperature (particularly from 15 to 30o C) may be explained
in terms of the following effects:
Journal of Engineering Volume 13 September 2007 Number 3
1784
1. A temperature increase usually increases the reaction rate which is the corrosion rate
and according to the Freundlich equation (Shrier (2), 1976):
n
O2Ck rate Corrosion (1)
Where k is rate constant of reaction, CO2 (concentration of oxygen) and n is order of
reaction. The rate constant (k) varying with temperature according to Arrhenius
equation (Shrier (2), 1976):
RTE
oekk
(2)
Where ko is constant, E is activation energy, R is universal gas constant and T is
temperature. Then from this formula ( RTE
oekk
) indicates that the k is increased
with increasing temperature and then the corrosion rate which is leads to increasing
the rate of zinc dissolutions.
2. Increasing seawater temperature leads to decreasing seawater viscosity with a
consequent increase in oxygen diffusivity according to stokes-Einstein equation
(Cussler, 1984 and Konsowa and El-Shazly, 2002):
constantT
D (3)
Where μ is the seawater viscosity and D is the diffusivity of the dissolved oxygen. As
a result of increasing the diffusivity of dissolved oxygen, the rate of mass transfer of
dissolved oxygen to the cathode surface increases according to the following
equation (Konsowa and El-Shazly, 2002):
22 O
d
Od CD
CkJ
(4)
With a consequence increase in the rate of zinc dissolution. Where kd is mass transfer
coefficient and J is mole flux of oxygen.
A. S. Yaro sacrificial anode cathodic protection
K. W. Hameed of low carbon steel in sea water
1785
3. The decreases in seawater viscosity with increasing temperature improve the
seawater conductivity with a consequent increase in corrosion current and the rate of
corrosion.
4. On the other hand, increase of temperature reduce the solubility of dissolved oxygen
with a subsequent decrease in the rate of oxygen diffusion to the cathode surface and
the rate of corrosion.
It seems that within the present range of temperature effects 1, 2 and 3 are predominating.
Flow rate effect
Figures 3, 5 and 7 show the effect of solution flow rate on the zinc dissolution with time, with
different temperatures and with different pH’s, respectively. It can be seen from figures 3, 5 and 7
that the dissolution rate of zinc increases with increasing the flow rate. This may be attributed to the
decrease in the thickness of hydrodynamic boundary layer and diffusion layer across which
dissolved oxygen diffuses to the tube wall of steel with consequent increase in the rate of oxygen
diffusion which is given by equation 4. Then the surface film resistance almost vanishes, oxygen
depolarization, the products of corrosion and protective film are continuously swept away and
continuous corrosion occurs. The flow rate of seawater may also caused erosion which combined
with electrochemical attack.
pH effect
Figures 4, 6 and 7 show the effect of pH on dissolution of zinc with time, with different
temperatures and with different flow rates, respectively. It can be seen from these figures that the
rate of zinc dissolution increases with decreasing of pH (particularly at range of pH 5 to 2). Within
the range of about 5 to 12 the corrosion rate of steel and hence dissolution rate of zinc is slightly
dependent of the pH, where it depends almost on how rapidly oxygen to the metal surface.
Although it was expected that at very high of pH value (12), the dissolution rate of zinc is much
reducing because the steel becomes increasingly passive in present of alkalies and dissolved
oxygen, but the nature of electrolyte (seawater) prevent that where chloride ions depassivate iron
even at high pH. Within the acidic region (pH<5) the ferrous oxide film (resulting from corrosion)
is dissolved, the surface pH falls and steel is more direct contact with environment. The increased
rate of reaction (corrosion) is then the sum of both an appreciable rate of hydrogen evolution and
oxygen depolarization.
Journal of Engineering Volume 13 September 2007 Number 3
1786
t ime (hr)
Zin
c c
on
su
mp
tio
n (
mg
/cm
2
)
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
T= 0 o
C
T= 15 o
C
T= 30 o
C
T= 45 o
C
Fig. 2, Zinc consumption with time for different temperatures at flow rate=600 lit/h and pH=8
time (hr)
Zin
c c
on
su
mp
tio
n (
mg
/cm2)
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Q = 5 lit/hr
Q = 300 lit/hr
Q = 600 lit/hr
Q = 900 lit/hr
A. S. Yaro sacrificial anode cathodic protection
K. W. Hameed of low carbon steel in sea water
1787
Fig. 3, Zinc consumption with time for different flow rate at temperature=30o C and pH=8
time (hr)
Zin
c c
on
su
mp
tio
n (
mg
/cm
2)
0
2
4
6
8
10
12
14
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
PH = 2
PH = 5
PH = 8
PH = 12
Fig. 4, Zinc consumption with time for different pH's at temperature 30o C and flow rate=600 lit/h
Journal of Engineering Volume 13 September 2007 Number 3
1788
Flowrate (lit/hr)
Zin
c c
on
su
mp
tio
n (
mg
/cm
2)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0 100 200 300 400 500 600 700 800 900 1000
T = 0 oC
T = 15 oC
T = 30oC
T = 45 oC
Fig. 5, Zinc consumption with flow rate for different temperatures at time=4 h and pH=5
pH
Zin
c c
on
su
mp
tio
n (
mg
/cm
2)
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
T= 0oC
T= 150C
T= 30oC
T= 45oC
Fig. 6, Zinc consumption with pH for different temperatures at time=4 h and flow rate=600 lit/h
A. S. Yaro sacrificial anode cathodic protection
K. W. Hameed of low carbon steel in sea water
1789
flowrate (lit/hr)
Zin
c c
on
su
mp
tio
n (
mg
/cm
2)
0
2
4
6
8
10
12
14
16
0 100 200 300 400 500 600 700 800 900 1000
PH = 2
PH = 5
PH = 8
PH = 12
Fig. 7, Zinc consumption with flow rate for different pH's at time=4 h and temperature=45o C
CONCLUSION
From the results obtained in the present work the following conclusion can be drawn: The
study of sacrificial anode cathodic protection of short steel tube using zinc strip extended axially in
the pipe revealed that under the present range of conditions of temperature, flow rate, pH of seawater
and protection time, the rate of zinc consumption increases with increasing temperature, flow rate and
time and with decreasing of pH. Zinc consumption during first hour is greater than during second
hour and so on. The zinc consumption with very low pH is very high and the cathodic protection
becomes unreliable.
REFERENCES
• Al-Mardy, S. M. (1999), "Design of fuzzy logic controller for the cathodic protection of
underground pipelines", M. Sc. Thesis
Bosich, F. J. (1970), "Corrosion prevention for practicing engineers", Barnes and Noble Inc.
Journal of Engineering Volume 13 September 2007 Number 3
1790
Brown, P. E. and Root, m. (2003), "The failure analysis experts", Cathodic Protection for
Deep Water Pipelines, Houston-Texas.
Cussler, E. L. (1984), "Mass transfer in fluid systems", Cambridge University press,
Cambridge, UK.
Davies, K. G. (2003), "Cathodic protection in practice". (internet site: www.npl.co.uk)
Halford, G. E. (1985), "Introduction to cathodic protection".
Jezmar, J. (2003), "Monitoring methods of cathodic protection".
Konsowa, A. H. and El-Shazly, A. A. (2002), Elsevier, Desalination, 153, pp 223-226.
Schweitzer, A. P. (1987), "What every engineer should know about corrosion".
Scully, J. C. (1975), "The fundamental of corrosion", 2nd
edition.
Shrier (1), L. L. (1976), "Corrosion 1, metal/environment reactions", Newnes-Butterworth.
Shrier (2), L. L. (1976), "Corrosion 2, corrosion control", Newnes-Butterworth.
Shrier (2), L. L. (1994), "Corrosion 2, corrosion control", 3rd
edition, Newnes-Butterworth.
Trethewey, K. R. and Chamberlain, J. (1996), "Corrosion for scientist and engineering", 2nd
edition.
Journal of Engineering Volume 13 September2006 Number 3
8371
Numerical Simulation of Heat Transfer Problem in Hot and Cold
Rolling Process
Ass.Prof. Dr. Ihsan Y. Hussain
Mech. Engr. Dept.
College of Engineering
University of Baghdad
Baghdad – Iraq
ABSTRACT
An efficient numerical model had been developed to model the thermal behaviour
of the rolling process. An Eulerian formulation was employed to minimize the number of grid
points required. The model is capable to calculate the temperature distribution, the heat penetration
depth, the convection heat transfer coefficient of cooling, the flow of metal through the roll gap, and
the heat generation by plastic deformation and friction. The roll is assumed to rotate at constant
speed, and the temperature variations are assumed to be cyclically steady state and localized with a
very thin layer near the surface. The Conventional Finite Difference (CFDM) based on cylindrical
coordinates was used to model the roll, and a Generalized Finite Difference Method (GFDM) with
non-orthogonal mesh was employed in the deformed strip region and the roll-strip interface area.
An upwind differencing scheme was selected to overcome the numerical instability resulting from
the high velocity ( high Peclet number ) involved in the rolling process. The equations of the strip
and roll are then coupled together and solved simultaneously. Both cold and hot rolling heat transfer
behaviours, velocity distribution, and heat generation by deformation and friction under typical
rolling conditions were presented to demonstrate the feasibility and capability of the developed
numerical model. It has been found that, while the strip is under deformation, the bulk temperature
inside the strip increases continuously; this is largely controlled by the deformation energy. On the
other hand, the strip surface temperature changes much more drastically and it is mainly controlled
by the friction heat and the roll temperature. The roll acts like a heat sink, because the coolant
heavily cools it. Thus, as soon as the strip hits the roll its surface temperature drops. Since
considerable friction and deformation heat are created along the interface and transferred from the
neighboring sub-layer, the surface temperature picks up rapidly. Finally, the results of the
temperature distribution for both cold and hot rolling and the heat generation by deformation and
friction obtained from the present study were compared with previous published work to verify the
validity of the numerical solution. Good acceptable agreements were obtained.
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9371
الخلاصـة
(. اسممتم مص صمم ر Rolling Process) أل رف مم نمممممممممو ع دمم ن لنم لممر التصممري المممرارن ل مممم ر إلمم التوصمم تمم
ل نمو ع ال ن القاب د . ل م ال ن( اللازم Mesh) ألشبكر( لتق د نقاط Eulerian Formulation) ر ال أو
( م امممم انتقممما الممممرار لمممما ت التبر ممم Heat Penetration Depthدمممما انتشمممار الممممرار )مسممماو توز مممت رلممماص الممممرار
(Cooling Heat Transfer Coefficient( لر ما الم مم مملا دم ممر ال رفممممم )Flow of Metal in Rolling )
Heatالممممرار المتولممم بسمممبو التشمممود ال ممم ل م ممم )
Generation by Plastic Deformation and Friction). تم افتمراث وبموص السمرد ال وران مر ل رف مممممم وا الت مراص التم
طر قمممممم ( د مممممم سممممممط المممممم رف .Very Thin Layerتطممممممرأ د مممممم رلمممممماص المممممممرار تكممممممو د مممممم ممممممر شممممممر ر قممممممر )
Polar( المبن م د م أسمالإ اامم او اص القطب م )Conventional Finite Difference Method) التق ل فرو اص المم
Coordinates . الفممرو مماص ب نممما اسممتم مص طر قمم ( مم اثكوممر ملا ممم والتمم اسممتم مص ا لمما توز ممت رلمماص المممرار ل رف ممممممم
تمم تكممو ف طمما مطمموط رلمماص المممرار ل منمماطا ال لنم لمم ( Generalized Finite Difference Methodالمممم الم مةممم )
لفممرو نسمما لقمم امت ممر ف طمما الم مم . تشممود التمم بالمنطقمم المتمو مم و (Non-orthogonal Mesh) الشممبكر ر ممر مت اممم
( ل ت و د دم اسسمتقرار ر ال م الناتلم مم Up-wind Differencing Schemeر ا )ل ل ل نقاط الموالط الصاد اص
لم م الم رفممممم مم م ما لت اتم الم ن تتمممنر دم مر ال رف م . ( High Peclet Numberو دم بك مص ال مال )السمر ال ال م أ
(Strip ) و( ال رف ممممممRoll( آن مما )Simultaneously.) درمممص م مما مممالت ال رف مم د مم البممار ود مم السممام تمممص راسمم
مممرار المتولمم بالتشممود واسمتكمما بمقتممم الشممروط الم و مم النمو ل مم ا ممما توز ممت السممر وتوز ممت ال التصممري المممرارن و نتمما
( أ مق ار رلر الممرار فم ام مر تمز ا Stripول ف ال راسر المال أوناء تشود الم )التوص ال ر. ا ر النمو ع ال ن ت
ام م أممرف فمد رلمر ممرار سمط الم م تت مر بصمورد أكومر بصورد مستمر وأ طا التشو ر تس طر د طا بصورد كب رد. مم ن
أوممارد والتمم تكممو طا ممر اسمتكمما و رلممر المممرار اممم المم رف تسمم طرد طا بصممورد ر سمم ر. أ التبر مم ال ممال المم ن ت ممرث لممر
فمد رلمر ممرار سمط (. م م دنم اصمط ا الم م بالم رفHeat Sinkالممرارن) ورالم رف سم كو لمر أومر كب مر أشمبر بمال
( أممافر إلم الممرار المنتق م Interfaceال رف تطبط. بسبو مرارت التشود واسمتكا ال تا تنشآ د طو سط المتلاملإ )
م ممرا نتمما توز مماص رلممر المممرار والمممرار أ ممم الطبقمم الملمماور ل م مم المشممود فممد رلممر مممرار سممط الم مم ترتفممت بسممرد .
Journal of Engineering Volume 13 September2006 Number 3
8371
المتول بسبو التشود واسمتكا الت ت التوص لطا ورنمص بنتما دمم مسمبا ل تمقما مم مم ف صمم المم ال م ن. لقم ولم
م المقارن ب النتا الممسوب م البم المال والنتا الممسوب ف دم مسبا. توافا ل
KEY WORDS: Thermal Behaviour, Hot and Cold Rolling , Roll and Strip, Velocity and
Temperature Distribution
INTRODUCTION
The process of plastically deforming metal by passing it between rolls is known as Rolling,
(Dieter 1986). It can be considered as one of the most important of manufacturing process.
Numerous investigations, numerical, analytical, and experimental have been carried out on rolling.
In hot or cold rolling the main objective is to decrease the thickness of the metal. Ordinarily little
increase in width occurs, so that decreases in thickness will result in an increase in length. As
predicted by (Lahoti and Altan 1975) the energy consumed in plastic deformation is transformed
into heat while a small portion of the energy is used up in deforming the crystal structure in
material. This heat generation coupled with heat transfer within the deforming material and to the
environment gives a temperature distribution in the deformed peace. As mentioned by (Karagiozis
and Lenard 1988), a ( 1) percent variation of the temperature may cause (10) percent change in
strength, which in turn will cause significant change in roll loads. As well as, the adequate cooling
of roll and the rolled products is of a considerable concern to rolls designers and operators.
Improper or insufficient cooling not only can lead to shorten roll life, due to thermal stresses, but it
can also significantly affect the shape or crown of the roll and result in buckled strips or bunted
edges. Considerable work has been done on modeling of the thermal behavior of rolling process.
(Johnson and Kudo 1960) used upper pound technique to predict the strip temperatures. (Lahoti et
al. 1998) used a two-dimensional finite difference model to investigate the transient strip and a
portion of the roll behavior. (Sheppard and Wright 1980) developed a finite difference technique
to predict the temperature profile during the rolling of the aluminum slab. (Zienkiewicz et al. 1981)
submitted a general formulation for coupled thermal flow of metal for extrusion and rolling by
using finite element method. (Patula 1981) with an Eulerian formulation attained a steady state
solution for a rotating roll subjected to prescribed surface heat input over one portion and
convective cooling over an other portion of the circumference. (Bryant and Heselton 1982) based
on the idea of “rotating line sources of heat” and the strip modeling based on the theory of heat
conduction in a “semi infinite body” they found that the knowledge of heat transfer mechanisms in
hot rolling was essential to the study of many areas of the process. (Bryant and Chiu 1982)
derived a simple model for the cyclic temperature transient in hot rolling work rolls. (Tseng
1984) investigated both the cold and hot rolling of steel by considering the strip and roll together.
(Tseng et al. 1990) developed an analytical model (Forier integral technique) to determine the
temperature profiles of the roll and strip simultaneously. (Remn 1998) used the Laplace and inverse
transform analytic technique to study the two dimensional unsteady thermal behavior of work rolls
in rolling process. (Chang 1998) used Finite difference formulations in the rolling direction and
analytical solutions were applied normal to this direction, making computational more efficient.
The experimental work done by (Karagiozis and Lenard 1988) shows the dependence of the
temperature distribution during hot rolling of a steel slab on the speed of rolling, reduction ratio and
initial temperature were investigated.
The purpose of the present study is to effectively analyze the thermal behavior of rolling
process for hot and cold rolling by considering the roll and strip simultaneously for two cases of
rolling conditions ( see Table 1 and Fig. 1 ) by using a suitable numerical methods.
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9379
MATHEMATICAL FORMULATION
The mathematical formulations of the problem will be presented in this article. The
following assumptions were made;
1. The strip and roll are long compared with the strip thickness therefore; axial heat conduction
can be neglected.
2. The steady state conditions are considered for both strip and roll.
3. Since tremendous rolling pressure builds up in the interface then, the surface roughness became
insignificant, and the film is very thin, on the order of micron, therefore, the thermal resistance of
the film can be neglected.
4. The constant friction coefficient was assumed along the interface.
5. No increase in width, so that the vertical compression of the metal is translated into an
elongation in the rolling direction.
Using an Eulerian description, the energy equation of the strip for a planer steady state
problem as shown in (Tseng 1984) is;
ssdss
sss cq
y
T
x
T
y
Tv
x
Tu
2
2
2
2
(1)
where (u) and (v) are the velocity component in (x and y) directions respectively which should
satisfy the equation of continuity, (α, ρ and c) are the thermal diffusivity, density and specific heat
respectively, (qd) is the rate of heat generation by deformation per unit volume and the subscript (s)
refers to the strip properties.
With respect to a fixed Eulerian reference frame, the governing partial differential equation
of the roll temperature (Tr) as shown in (Tseng 1990) is;
2
2
22
211
rrrr
r
T
rr
T
rr
TT
(2)
where (r) and (θ) are the cylindrical coordinates; (ω) is the roll angular velocity; and the subscript
(r) refers to the roll properties.
The concept of the thermal layer has also been applied to a numerical analysis by (Tseng
1984) and in the present study, it has been improved computational accuracy.
According to (Tseng 1984), ( R ) can be found as a function of the Peclet number
( rRPe 2 ). Alternately, following (Patula 1981), showed that ( PeR 24.4 ), when
( Pe >>0), a condition satisfied in most commercial strip rolling.
Based on a numerical study of (Tseng 1984) the;
PeR 7 (3)
is large enough for the present numerical model.
For rolling situations involving high speeds, the penetration would be significantly less
where ( rRPe 2 ). Conversely, for lower rotational speeds, the penetration would be greater.
Journal of Engineering Volume 13 September2006 Number 3
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In the present study (second case) and as reported by (Tseng 1990), the mean film
coefficient of the water-cooling spray is about (3.4 W/cm2.oC) over about 30 degrees of the roll
circumference. The secondary cooling produced by water puddling varies from (0.28 to 0.85
W/cm2.oC) as shown in Fig. 2. The two peak squares represent the entry and exit cooling. Puddling
covers the remaining area.
As well as for the first case, the convection heat transfer for water cooling spray varies from
(0.85 W/cm2.oC to 3.4 W/cm
2.oC) along the roll. The heat transfers coefficient as presented in Fig.
(4) varies as half sine curve to simulate both the entry and exit cooling. Then, from Fig.3. the heat
transfer coefficient can be written as;
)sin(55.285.0)( H (4)
Then, the two cases of water-cooling spray, Figs. 2 and 3 are considered in the present study
to simulate the entry and exit cooling during the rolling.
The flow of metal under the arc of contact is determined by assuming that the volume flow
rate through any vertical section is constant. A metal strip with a thickness (to) enters the bite at the
entrance plane (XX) with velocity (uo). It passes through the bite and leaves the exit plane (YY)
with a reduction thickness (tf ) and velocity (uf ) as shown in Fig. 4.
Since equal volumes of metal must pass at any vertical section, then;
ffoo uWtWtuuWt (5)
where (W) is the width of strip; (u) is the velocity at any thickness (t) intermediate between (to) and
(tf).
At only one point along the arc of contact between the roll and strip is the surface velocity
of the roll (Vr) equal the velocity of the strip. This point is called the neutral point or no-slip point
and indicated in Fig. 4 by point (N).
If large back tension or heavy draught is applied, the neutral point shifts toward the exit
plane and the metal will slip on the roll surface, this (back tension) condition is considered in the
present study.
There are two components of velocity, one of these in (x) direction is denoted by (u) and the
other in (y) direction is denoted by (v). Recall eq. (5), then;
rnffoo VWtWtuuWtuWt (6)
Thus;
rn Vt
tu
(7)
When the equation of continuity is satisfied, then;
0
y
v
x
u (8)
Substitute eq. (7) in eq. (8), thus;
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9377
0
y
vV
t
t
xr
n (9)
After some arrangement the eq. (9) gives;
dydx
dtV
t
tdv r
n
2 (10)
After integration eq. (10) becomes;
ydx
dtV
t
tv r
n
2 (11)
where ( dxdt ) is the slope of the arc of contact at any (x).
In the present study for the second case, the deformation heat is distributed in the strip in
proportion to the local effective strain rate and same as that distributed in (Tseng 1984);
22 )()( yxeff (12)
Or;
22
h
v
x
ueff (13)
In general, the deformation heat is proportional to both the strain rate and the flow stress,
then;
flowseffdQ ) (14)
where (Qd) is the deformation heat generation (Kw); ( eff ) is the strip effective strain rate (1/s)
and ( s )flow ) is the strip flow stress (N/cm2).
The distribution assumed in eq. (14) implies that the flow stress variation is small compared
with very large strain rate as predicted by (Zienkiewicz et al. 1981) and (Tseng 1984), then;
effdQ (15)
Substitute eq. (8) with (y=h) and (7) in eq. (13), then;
r
neff V
h
h
x2 (16)
After differentiation the eq. (16), then;
dx
dhV
h
hr
neff 2
2 (17)
Journal of Engineering Volume 13 September2006 Number 3
8377
where ( hn) is the half thickness of the strip at the neutral point.
In the present study for the second case, the friction heat is distributed along the interface in
proportion to the magnitude of the slip (relative velocity) between the roll and the strip as shown in
Fig. 5, then;
slipfr VQ (18)
where ( frQ ) is the heat generation by friction (kW).
It is well known from Fig. 8 that the slip velocity is;
22 vuVV rslip (19)
As well as, the heat generated by deformation and friction for the first case study is assumed
to be uniformly distributed in the bite and interface.
The input data of the heat generation by deformation and friction will be obtained from
direct measurement of the power (Table 1) and it is considered in the present study. As shown by (Tseng 1984), before entering and after exit the strip into and from the roll
bite, the strip loses heat to the coolant by convection, see Fig. 6, then;
)(
TTH
n
Tk s
ss (20)
In the present study and in (Dieter 1986 and Tseng 1984), since the strip velocity is high,
the conduction term, (22 xTs ) becomes small in comparison with the convection term,
( xTu s ). Thus the billet temperature should be the initial strip temperature (To).
The boundary condition as shown by (Tseng 1984) at some distance downstream from the
exit contact point may be assumed to be;
0 xTs (21)
As showed by (Tseng 1984) because of the symmetry, the lower horizontal boundary
having;
0 yTs (22)
The boundary condition for the roll circumference is;
TRTH
r
RTk r
rr
,
, (23)
where ( )(H ) is the heat transfer coefficient explained previously.
Since the roll is rotate rapidly, and all temperatures vary within a very thin layer near the
surface, only a thin layer needs to be modeled. The interior boundary condition as shown by (Tseng
1984) becomes;
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9371
0),( rRTr (24)
The strip is assumed to be in contact the roll and each moves relative to the other, creating
the friction heat along the interface. Mathematically, as shown by (Tseng 1984) this boundary
condition may be expressed as;
0
fr
b
rr
b
ss q
n
Tk
n
Tk (25)
where ( n ) represents differentiation along the normal of the boundary (positive outward); see
Fig. 9; and (qfr) is the friction heat generated along the interface and the subscript (b) refers to the
boundary. All special derivatives for the points at the interface must be formulated using points
located in their respective sides as follows;
21 qqn
Tk
b
ss
(26)
Then, from Fig. 7;
b
ss
o
b
ss
o
x
Tk
q
y
Tk
q
12 cos,sin
(27)
Substituting eq. (27) in eq. (26), thus;
o
b
sso
b
ss
b
ss
y
Tk
x
Tk
n
Tk sincos
(28)
where the angle ( o ) specifies the direction as shown in Fig. 7.
Because the two bodies are in intimate contact, temperature equality at the interface is also
assumed;
brbs TT (29)
Replacing bs nT by the directional derivatives eq. (28) and
br nT by
br rT and from Fig. 7, eq. (25) becomes;
0sincos
s
fr
b
r
s
ro
b
so
b
s
k
q
r
T
k
k
y
T
x
T (30)
Journal of Engineering Volume 13 September2006 Number 3
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`NUMERICAL SOLUTION
In this article, the task of constructing the numerical method for solving the governing
partial differential eqs. (1) and (2).
The essence of GFDM is its ability to obtain the needed derivative expression at a given
point as a function of arbitrarily located neighboring points. As reported by (Tseng 1984) for any
sufficiently differentiable function, T(x,y), in a given domain has Taylor series expansion about a
point (xo,yo) up to second order terms can be written as;
o
sii
o
si
o
si
o
si
o
sisosi
yx
Tnm
y
Tn
x
Tm
y
Tn
x
TmTT
2
2
22
2
22
22 (31)
where (Ti=T(xi,yi), To=T(xo,yo), mi=xi-xo)and (ni=yi-yo). Five independent equations, similar to eq.
(31), can be obtained by using five arbitrarily located neighboring points (xi,yi), i=1,…,5, as shown
in Fig. 8.
If the first special derivatives ( xTs … yxTs 2) at point (xo,yo) can computed in
terms of the functional values at five neighboring points, see Fig. 8. In matrix form, as mentioned in
(Tseng 1984);
5525
2555
4424
2444
3323
2333
2222
2222
1121
2111
22
22
22
22
22
nmnmnm
nmnmnm
nmnmnm
nmnmnm
nmnmnm
yxT
yT
xT
yT
xT
s
s
s
s
s
2
22
22=
sos
sos
sos
sos
sos
TT
TT
TT
TT
TT
5
4
3
2
1
(32)
Or;
sosisjji TTDTA , i , j=1,2,…,5 (33)
Inverse of the matrix [Ai.j] leads to;
sosijisi TTBDT , i , j=1,2,…,5 (34)
where [Bi,j] is the inverse of [Ai.j]. Rearranging eq. (34), then;
][ , sjjisi TBDT i =1,…,5, j=0,…,5 (35)
where;
5
1jijio BB (36)
Finally the special derivatives at point (xo,yo) can be found as reported by (Tseng 1984) as;
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9373
5
1oj
jsj
o
s TBx
T (37a)
5
2oj
jsj
o
s TBy
T (37b)
5
32
2
ojjsj
o
s TBx
T (37c)
5
42
2
ojjsj
o
s TBy
T (37d)
Substituting the eqs. (36), (37a), (37b), (37c) and (37d) in the strip governing eq. (1), an
algebraic approximation for each internal point was as reported by (Tseng 1984);
oosoooo
j sjjsjsjojossd
soBBBvBu
TBBBvBucqT
4321
5
1 4321
(38)
For the boundary points, spatial care is required. Substituting eq. (30) with
rTTrT rrobr and ( jo rrr ) into eq. (31), from eq. (30) after the final
substitution, eliminate os xT or
os yT . If ( 0o or ), keep os xT and find;
ii
o
si
o
si
o
si
o
siisoisi nm
yx
Tn
y
Tm
x
Tna
x
TnamTnaT
22
2
22
2
2
3212
1
2
11
(39)
where;
os
r
rk
ka
sin1 (40)
oa cot2 (41)
frr
r
os
qTr
k
ka
sin
13 (42)
As reported by (Tseng 1984), upon providing four arbitrary selecting neighboring points,
Fig. 9, four independent equations similar to eq. (39) can be obtained by following the procedure
similar to that for treating the internal points, then;
soisjij TfDTD ][ i, j=1,……,4 (43)
Journal of Engineering Volume 13 September2006 Number 3
8371
where;
i
iii
na
namD
1
21
1
,
i
ii
na
mD
1
2
212
, i
ii
na
nD
1
2
312
i
iii
na
nmD
14
1 ,
i
isii
na
naTf
1
3
1
and DTsi is column matrixes containing the four derivatives of eq. (39).
Again, inversion of [Dij] leads to;
][ jijsi fEDT i=1,…,4, j=0…, 4 (44)
where [Eij] is the inverse of [Dij], fo=Tso , and
4
1j ijio EE . Thus, the special derivatives at the
boundary points, (xo,yo) become;
4
01
jjj
o
s fEx
T (45a)
4
022
2
jjj
o
s fEx
T (45b)
And;
4
032
2
jjj
o
s fEy
T (45c)
Substituting eq. (45a) in eq. (28) and determining os yT , then;
4
03112
jojj
o
s afafEay
T (45d)
Substituting the eqs. (45a), (45b), (45c) and (45d) into strip governing eq. (1), an algebraic
relationship for the boundary point (xo,yo) was as reported by (Tseng 1984);
oosoooo
ssdojjsjj joojjs
soEEvaEvau
cqvananaTEvauEET
32112
3134
1 1232 1
(46)
Upwind scheme was employed to achieve numerical stability, then;
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9371
o
si
o
sisosi
y
Tn
x
TmTT
(47)
Two simultaneous equations were obtained by using two neighboring points (1 and 2). After
rearranging these two equations into matrix form, then;
sjj
j
o
sTF
x
T
2
01 (48a)
sj
j
j
o
sTF
y
T
2
0
2 (48b)
Substitute the eqs. (60a) and (60b), and those in eqs. (49c) and (49d) to the strip governing
eq. (1), the first upwind GFDM equation for each internal point was as reported by (Tseng 1984);
oosoooo
j j sjjjssjjsjsjoijossd
soBBFvFu
TBBTBBFvFucqT
4321
2
1
5
3 43432
(49)
For the typical boundary condition described in eq. (30), and using the same notations as in
eqs. (39) and (43) for (ai and fj ), respectively, point (1) is again to be an upwind point, Fig. 9, then;
)0(1 321
termsallna
x
TnamTnaT i
o
siisoisi (50)
Or;
1
0jjj
o
fGx
T (51)
And;
1
0312
jojj
o
afafGay
T (52)
where 121111 1 namnaGG o .
Substitute the eqs. (51), (52), (45b) and (45c) into strip governing eq. (1), an upwind
GFDM relationship for a boundary point (xo,yo) was as reported by (Tseng 1984);
oosoooo
j ssdooojjjs
soEEvaGvau
cqvafGvaufEET
3212
4
1 311232
(53)
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8371
The roll governing eq. (2), is approximated by using second order central differencing for
the conduction terms (right side) and first order up wind differencing for the convection terms (left
side).
The temperature profile becomes identical in a plot of normalized temperature,
oror qTTHT )(* , against, Rrr *
, which will certainly simplify further parametric study
and high accuracy.
Then, in dimensionless form the roll governing eq. (2) becomes;
*
2
*2
2
*2
2**
*
*
11 rrr
o
r
o
TPe
r
TT
rr
T
r (54)
where the superscript (*) refers to the dimensionless quantity.
By using four arbitrary located neighboring points as shown in Fig.10, then the roll
governing eq. (2) becomes;
*1
*
2
*1
**3
**
*2
*4
**
*2
**4
2
21
2
1222
rrorror
o
rr
o
rror TTPe
TTT
rr
TT
rr
TTT
(55)
Rearranging the above equation, an algebraic approximation for roll internal nodes is;
rorrrrrrrrro aTaTaTaTaT *44
*33
*22
*11
* (56)
where;
2*3**2*
22*1
2
1,
2
11,
2
1
o
r
o
r
o
r
ra
rrra
Pe
ra
**2*4
2
11
rrr
a
o
r
,
2*2*
1
2
12
rr
Pea
o
ro
RESULTS AND DISCUSSION
This article presents the numerical results of the present work, besides, a verification of the
computational model will also be made.
Fig. 11 show the horizontal component of velocity for the first case study. In the billet
region, the strip has velocity (u) only, i.e., (v=0). Since equal volumes of metal must pass at any
vertical section through the roll gap and the vertical elements remain undistorted (no increase in
width), eq. (13) requires that the exit velocity must be greater than entrance velocity, therefore, the
velocity (u) of the strip must be steadily increased from entrance to exit. the exit product region
having (u) velocity only, i.e., (v=0).
The vertical component of velocity (v) for the first case study and for different lines are
shown in Fig.12. In the billet, the streamlines are horizontal and having horizontal component of
velocity (u) only and (v=0). After entering the bite the streamlines have curved shapes and the
slopes of these curves gradually decrease from entrance to exit. Then the velocity (v) gradually
diminishes from entrance to exit as shown in eq. (11) and in Figs.12. The horizontal and vertical
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9319
components of the velocity for the second case study are similar to those in the first case but the
different in the elevations
As mentioned by many authors such as (Tseng 1984) and (Lahoti et al. 1978), rolling a
mild steel as shown in Table 1, consume about (90) percent of the total power in the deformation of
the strip and in the friction loss at the interface. Moreover, according to (Zienkiewicz et al. 1981)
and (Tseng 1984) who measured both the plastic work and the temperature rise in a tensile
experiment. It was found that for steels, copper and aluminum, the heat rise represents (86.5, 90.5-
92 and 95) percent, respectively, of the deformation energy which is converted into heat.
In the present study, this (90) percent estimated is used, i.e., that (6.5) percent of (90)
percent of total power is dissipated as friction heat along the interface.
The resulting values of the heat generation by deformation and friction were summarized in
the Table 2 for the first and second case study. In the first case, the heat generation by deformation
and friction are distributed. In the second case, the heat generation by deformation is then
distributed to the strip in proportion to the local effective strain rate as shown in eqs. (15) and (17)
and Fig.13. Note that the highest strip deformation (strain rate) occurs near the bite entry and
diminishes monotonically toward the end of the bite as shown in Fig.13. Thus, the highest heat
generation by deformation occurs near the bite entry too and diminishes monotonically toward the
end of the bite as shown in eq. (19) and Fig. 13.
Also, the strip is to be moved relative to the roll creating friction heat along the interface as
recorded in eq. (19), i.e., the friction heat is then distributed in proportion to the (relative velocity
between the strip and roll as recorded in eq. (18) and in Fig.14. The maximum slip occurs at the
first point of contact at the interface because the roll draws the thick strip into the bite. Then, the
slip velocity decreases gradually until sticking at the final point of contact as shown in Fig. 14.
Figs.15 and 16, indicate that the roll temperature variations are limited within a very thin
layer, about (1) percent of the radius, which consistent with the associated boundary condition eq.
(24). The surface temperature rapidly increases at the bite due to great heat generated by the friction
and transferred from the strip. As the roll leaves the bite, the roll surface temperature immediately
decreases due to heat convected to the coolant and heat conducted into the immediate sub surface
layer.
As well as, as shown in Figs. 15 and 16 the different in temperatures between the final and
initial points of contact for the first case study is less than for the second case. This means, using
several small coolant sprays (second case) is more efficient than one large spray (first case).
Figs. 17 and 18 indicate that while the strip is under deformation, the bulk temperatures
inside the strip increase continuously; this is largely controlled by the deformation energy. On the
other hand, the strip surface temperature changes much more drastically and it is mainly controlled
by the friction heat and the roll temperature.
The coolant heavily cools the roll; it acts like a heat sink. Thus, as soon as the strip hits the
roll its surface temperature drops as shown in Figs. 17 and 18. Since considerable friction and
deformation heat are created along the interface and transferred from the neighboring sub layer, the
surface temperature picks up rapidly.
Beyond the bite, Figs.17 and 18, the strip temperature tends to be uniform. In this region,
the heat convected to the air has been assumed to be negligible. For high-speed rolling (rather than
the considered limits), the product temperature behaves parabolically rather than elliptically as
implied by eq. (1). In other words, the boundary conditions that are assumed in the product should
not have a noticeable effect on the bite region.
The interface heat fluxes results for uniform and non-uniform heat generation distributions
are shown in Figs.19 and 20. At the initial contact stage, as anticipated, a very large amount of heat
is transferred to the roll. In fact, the roll surface temperature is about (25 oC and 11.0362
oC) lower
than that of the strip as shown in Figs. 15, 16, 17 and 18.
To satisfy the boundary condition eq. (29), a step change of surface temperatures are
expected to occur at the initial contact point (x=0). The induced heat flux to the roll at (x=0) as
Journal of Engineering Volume 13 September2006 Number 3
8371
shown in Figs. 19 and 20, also ensure the above findings that a large amount of heat is transferred
to the roll from strip and the interface friction at the initial contact stage.
It is believed that in the previous studies, the strip initial temperatures were close to that of
the roll. Therefore, the strip is not expected to have a temperature drop at the initial contact stage.
However, it is note worthy that at very high rolling speeds, measuring the local temperature change
in the bite could be a big challenge as mentioned previously in (Tseng 1984).
In hot rolling, the strip is normally rolled at elevated temperatures at which re-crystallization
proceeds faster than work hardening. In addition, the hot strip is generally rolled at thicker gages
and lower speed than that of the cold strip.
The gages specified in the first case are still suitable for hot rolling. Two focuses are
considered. The first focuses on the effect of changing the entering temperature to (900 oC). The
second, changing velocity by slowing the roll speed from (1146.6 to 573.3 cm/s). The other
operating conditions are similar to those discussed for cold rolling.
Fig. 21 depicts the roll temperature distribution for the two hot rolling cases consider
(Vr=1146.6 and 573.3 cm/s). A comparison of Fig.21 with Fig.17 indicates that the temperature
profile between the hot and cold rolling is mainly in magnitude but not in shape.
Both the interface heat flux and speed govern the temperature magnitude. As shown in Fig.
22 at speed of (1146.6 cm/s), the heat flux increases about four times for the hot to cold rolling. The
corresponding increase of temperature is also found to be about four times too as shown in Fig. 21.
Fig. 21 shows except in the bite region, the roll temperature is reducing about (15) percent
with the speed slowed to (50) percent, and the different in the bite region is much smaller and the
maximum temperature occurs at the end the arc of contact. For example, the corresponding
decrease of the peak temperature is less than (2) percent. The temperature decrease due to slowing
the speed is mainly due to decrease of the heat flux Fig. 22.
Figs. 21 and 23 also show that near the bite, very large temperature variations are within a
very thin layer. The layer thickness (δ), consistent with the previous finding, is dependent on the
speed, or more precisely, the roll Peclet number as shown in eq. (3).
The strip temperatures for the two hot rolling cases are presented in Fig. 24. In the bite
region, the strip temperature, similar to the roll temperatures, is not noticeably affected by changing
the speed within the range consider. In the down stream region (x>xo), the strip center temperature
drops faster in the slower strip. By contrast, the surface temperatures are not sensitive to the speeds
considered. This figure also indicates the temperature drop in the initial contact stage is much large
than its counterpart for the clod strip, as shown in Fig. 24. When the strip entry temperature rises
from (65.6 oC) to (900
oC) from cold to hot rolling, the temperature drop increases approximately
from (25 oC) to (649
oC), reflecting the great increase in the temperature different between the strip
and roll a head of the bite.
The shape of the heat input distribution to the roll (qr) governs the roll and strip
temperatures in the roll gap region. As shown in Figs.18 and 23 with a parabolic distribution of (qr)
of the second case study, the location of the maximum temperature shifts to the interior of the arc of
contact (heating zone). Although, the cumulative energy input is still increasing beyond (Φ/2), the
flux is decreasing, yet the effect of the type of heat distribution on the temperature distribution
away from the roll gap should be minimal as shown in Figs 21 and 23.
CONCLUDING REMARKS
1. While the heat generation by deformation occurs in the strip or by friction at the strip-roll
interface and the heat removal is at the roll surface, then, both strip and roll should be considered
together and solved simultaneously.
2. The highest heat generation by the deformation and friction occurs at the entrance to the bite
and diminishes gradually toward the end of the bite.
3. The results show that the extremely large temperature drop at the interface and large
temperature variation in both roll and strip are found. Such high temperature variations could
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9317
create very large (δ) thermal stresses within the thin layer and this stresses lead to the roll wear or
roll failure, then proper control of this stresses could significantly extend the roll life.
4. Several small coolant sprays (second case) are more efficient than one large spray (first case).
5. The temperature decreases due to slowing the roll speed. This is mainly due to decrease the
total input power that led to decrease of the heat flux at the interface.
6. The shape of the heat input distribution (uniform or parabolic heat input) to the roll governs the
location of the peak temperature.
Finally, the comparison of the present results with published findings by (Tseng 1984)
shows that the computational scheme used is effective and reliable. However, it is believed that the
greatest uncertainty in analysis will arise not from the numerical scheme, but from the input data,
in particular, the friction energy, the location of the neutral point (or the forward slip), and the heat
transfer coefficient of coolant.
Table 1 : Operational Parameters for the First and Second Case Studies.
Operational Parameters First Case, for Coil 45,
Tseng 1984.
Second Case, for Coil 32,
Tseng 1984
Strip Material. Mild Steel. Mild Steel.
Roll Material. Cast Steel. Cast Steel.
Coolant. Water. Water. Entry Gauge. 0.15 cm. 0.085 cm.
Exit Gauge. 0.114 cm. 0.057 cm.
Roll Speed. 1146.6 cm/s. 1219 cm/s.
Forward Slip. 0 0
Strip Width. 63.5 cm. 81.3 cm.
Roll Diameter. 50.8 cm. 50.8 cm.
Total Input Energy. 3694 kW. 3340 kW.
Strip Entry Temperature. 65.6 oC. 65.6
oC.
Both the roll and the strip have the following thermal properties: -
Thermal conductivity (kr, ks); 0 .4578 W/cm.oC
Thermal diffusivity ( sr , ); 0.1267 cm2/s
Table 2 : Amounts of the Heat Generation by Deformation and Friction.
Case Number Heat Generation by Plastic
Deformation Qd (kW)
Heat Generation by Friction frQ (kW)
8st 1771 891
1nd
1119 183
Journal of Engineering Volume 13 September2006 Number 3
8377
Fig. 1: Typical Arrangement of Rolls for Rolling Process.
Billet Product
Upper Roll
Lower Roll
Back tension.
Fig. 2: Cooling Heat Transfer Coefficient, Tseng 1984
200 240 280 320 360
ANGULAR LOCATION, (degree)
0
1
2
3
4
HE
AT
TR
AN
SF
ER
CO
EF
FIC
IEN
T
h
( ),
(W
/cm
. C
)
0 40 80 120 160 200
o
2
0
H
Bite
Fig. 3: Heat Transfer Coefficient that Varies as A Half Sine Curve,Patula 1981
Bite
160 200 240 280 320 360
ANGULAR LOCATION, (degree)
0
1
2
3
4
HE
AT
TR
AN
SF
ER
CO
EF
FIC
IEN
T
h
( )
, (
W/c
m . C
)
0 40 80 120 160
0
o2
H
θ r
ω
x
y
F P
r
Y
Y
N
X
X
xo
to uo tf uf
R
Fig. 4: Forces Acting During Rolling,Dieter 1986
Φ Φ
Fig. 5: The Relative Velocity between the Roll and Strip.
22s vuV
u
v
Roll
Bite
Interface R
h(x)
Vr
sss TTkHn/T
0n/Tr
Axis of symmetry
0n/Ts
Fig. 6: Typical Boundary Conditions for Strip, Roll and
Interface Region, Tseng 1984
r qfr
R-δ θ
0n/Ts
os TT
rrr TTkHn/T
sss TTkHn/T
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9311
Fig. 11: Horizontal Component of Strip Velocity.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
DISTANCE FROM ENTRY POINT, x(cm).
800
900
1000
1100
1200
HO
RIZ
ON
TA
L V
EL
OC
ITY
, u
(c
m/s
).
y/h=0.0-1.0
First Case.
Vr =1146.6 cm/s
Bite
x
Tk s
s
o
n
Tk r
r
x
n y
o
Boundary o
y
Tk s
s
q1
q2
o
Fig. 7: The Components of Interface Heat Flux.
qfr 1 2
3 0
4 5
Fig. 8: The Grid Arrangement for the Strip
Internal Nodes, Tseng 1984
Fig. 9: The Grid Arrangement for the Strip Boundary
(Interface) Nodes, Tseng1984
1 2
3
0
4
x
y n
Boundary
βo
r
θ
ω
3 o 4
1 2
θ
r
δ Roll
R
Fig. 10: Neighboring Points Arrangement for Roll Internal Nodes.
θ
Fig. 12: Vertical Component of Strip Velocity.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
DISTANCE FROM ENTRY POINT, x (cm).
-40
-30
-20
-10
0
VE
RT
ICA
L V
EL
OC
ITY
, v
(c
m/s
).
y/h=0.0 (Centre)
y/h=0.3334
y/h=1.0(Surface)
y/h=0.6667
First Case.
Vr =1146.6 cm/s
Bite
Journal of Engineering Volume 13 September2006 Number 3
8371
Fig. 13: Effective Strain Rate and Heat Generation by Deformation.
0.0 0.2 0.4 0.6 0.8 1.0
NORMALIZED BITE LOCATION, x/x .
0
200
400
600
800
1000
HE
AT
GE
NE
RA
TIO
N B
Y D
EF
OR
MA
TIO
N q
(k
W/c
m
).
0
200
400
600
800
1000
EF
FE
CT
IVE
ST
RA
IN R
AT
E, (1
/s).
d3
o
eff
Second Case.
Deformation Heat.
Effective Strain Rate.
y/h=0.0-1.0
Fig. 14: Slip Velocity and Heat Generation by Friction.
o
At the Interface.
0.0 0.2 0.4 0.6 0.8 1.0
NORMALIZED BITE LOCATION, x/x .
0
2
4
6
8
10
HE
AT
GE
NE
RA
TIO
N B
Y F
RIC
TIO
N q
(
kW
/cm
).
0
100
200
300
400
500
SL
IP V
EL
OC
ITY
, V (c
m/s
).
fr2
slip
Second Case.
Slip Velocity.
Friction Heat.
Fig. 15: Heat Transfer Coefficient and Roll Temperature for Cold Rolling Case.
0 40 80 120 160
0
20
40
60
80
100
120
RO
LL
TE
MP
ER
AT
UR
E,
( C
).
160 200 240 280 320 360
ANGULAR LOCATION, (degree).
0
0
1
2
3
4
HE
AT
TR
AN
SF
ER
CO
EF
FIC
IEN
T H
( ), (W/c
m . C
).
r/R=0.0-0.99 (core) r/R=1 (surface)
Bite
2
o
o
Entry Cooling Exit Cooling
1st. Case Vr=1146.6 cm/s
Roll Temp.
Heat Tran. Ceff.
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9313
Fig. 16: Heat Transfer Coefficient and Roll Temperature for Cold Rolling Case.
0 40 80 120 160 200
0
20
40
60
80
100
120
140
RO
LL
TE
MP
ER
AT
UR
E,
( C
).
200 240 280 320 360
ANGULAR LOCATION, (degree).
0
0
1
2
3
4H
EA
T T
RA
NS
FE
R C
OE
FF
ICIE
NT
H( ), (W
/cm
. C).
o
2o
r/R=0.0-0.99 (Core) r/R=1.0 (Surface)
Bite
2nd Case Vr=1219 cm/s
Roll Temp.
Heat Tran. Coeff.
Fig. 18: Comparison of the Strip Temperature for Cold Rolling.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
DISTANCE FROM ENTRY POINT, x(cm).
0
40
80
120
160
200
ST
RIP
TE
MP
ER
AT
UR
E (
C
).o
y/h=0 (Centre)
y/h=1 (Surface)
y/h=0.934y/h=0.866
Bite
2nd Case Vr=1219 cm/s
Refrence [15].
Present Study.
Fig. 19: Distributions of the Interface Heat Flux.
0.0 0.2 0.4 0.6 0.8 1.0
NORMALIZED BITE LOCATION, x/x .
0
5
10
15
20
25
30
HE
AT
FL
UX
, (k
W/c
m
).2
o
First Case.
Friction Heat.
Heat Transferred to Roll.
Heat Transferred from Strip.
Fig. 20: Distributions of the Interface Heat Flux.
0.0 0.2 0.4 0.6 0.8 1.0
NORMALIZED BITE LOCATION, x/x .
0
2
4
6
8
10
12
HE
AT
FL
UX
, (k
W/c
m
).2
o
Second Case.
Friction Heat.
Heat Transferred to Roll.
Heat Transferred From Strip.
Fig. 17: Comparison of the Strip Temperature for Cold Rolling.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
DISTANCE FROM ENTRY POINT, x(cm) .
40
60
80
100
120
140
160
ST
RIP
TE
MP
ER
AT
UR
E,
( C
).o
y/h=0.0 (Centre)
y/h=0.866y/h=0.934
y/h=1.0 (Surface)
Bite
1st. Case Vr=1146.6 cm/s
Refrence [16].
Present Study.
11
Journal of Engineering Volume 13 September2006 Number 3
8371
REFERENCES
Bryant, G. F. and Heselton, M. O., ”Roll Gap Temperature Models for Hot Mills,” Metals
Technology, 1982, Vol. 9, pp. 469-476.
Bryant, G. F. and Chiu, T. S. L., “Simplified Roll Temperature Model (Spray Cooling and Stress
Effects),” Metals Technology, 1982, Vol. 9, pp. 485-492.
Chang, D. F., “ An Efficient Way of Calculating Temperatures in the Strip Rolling process,” ASME,
Journal of Manufacturing Science and Engineering, 1998, Vol. 120, pp. 93-100.
Dieter, “Metals Metallurgy,” Mc Graw-George, E-Hill Book, Dieter company, Third addition,
1986.
Johnson, W. and Kudo, H., “The Use of Upper-Bound Solutions for the Determination of
Temperature Distributions in Fast Hot Rolling and Axi-Symmetric Extrusion Process,”
International Journal of Mechanical Science, 1960, Vol. 1, PP. 175-191.
Fig. 21: Roll Temperature near the Bite for Hot Rolling Cases.
-1 0 1 2 3 4 5 6 7
ANGULAR LOCATION, (degree).
0
100
200
300
400
500
RO
LL
TE
MP
ER
AT
UR
E,
( C
).
359
o
r/R=1.0 (Surface)
r/R=0.9995
r/R=0.993
r/R=0.9990
Bite
Roll Speed (cm/s).
Vr=1146.6
Vr=573.3
Fig. 22: Distributions of the Interface Heat Flux to Roll.
0.0 0.2 0.4 0.6 0.8 1.0
NORMALIZED BITE LOCATION, x/x .
0
20
40
60
80
100
120
HE
AT
TR
AN
SF
ER
ED
TO
RO
LL
, (k
W/c
m
).2
o
Roll Speed (cm/s)
1146.6 (Hot Rolling).
573.3 (Hot Rolling).
1146.6 (Cold Rolling).
Fig. 23: Roll Temperature near the Bite for Cold Rolling Case.
-1 0 1 2 3 4 5 6 7
ANGULAR LOCATION, (degree).
40
60
80
100
120
140
RO
LL
TE
MP
ER
AT
UR
E,
( C
).
359
o
r/R=1 (surface)
r/R=0.995
r/R=0.9990
Bite
Second Case.
Vr=1219 cm/s
Fig. 24) Strip Temperature for Hot Rolling Cases.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
DISTANCE FROM ENTRY POINT, x (cm).
0
200
400
600
800
1000
ST
RIP
TE
MP
ER
AT
UR
E,
( C
).o
Strip Exit Speed.
573.3 cm/s
1146.6 cm/s
y/h=0.0 (Centre)
y/h=0.866
y/h=0.934
y/h=1.0 (Surface)
Bite
I. Y. Hussain Numerical Simulation of Heat Transfer
Broblem in Hot and cold rolling Process
9311
Karagiozis, A. N. and Lenard, J. G., ”Temperature Distribution in A Slab During Hot Rolling,”
ASME, Journal of Engineering Materials and Technology, 1988, Vol. 110, pp.17-21.
Lahoti, G. D. and Altan, T., “Predication of the Temperature Distribution in Axi-Symmetric
Compression and Torsion,” ASME, Journal of Engineering Materials and Technology, 1975, Vol.
97, pp.113-120.
Lahoti, G. D., Shah S. N. and Altan, T., ”Computer Aided Heat Transfer Analysis of the
Deformation and Temperatures in Strip Rolling,” ASME, Journal of Engineering for Industry,
1978, Vol. 100, pp. 159-166.
Patula, E. T., “ Steady-State Temperature Distribution in A Rotating Roll Subjected to Surface Heat
Fluxes and Convective Cooling,” ASME, Journal of Heat Transfer, 1981, Vol. 103, pp. 36-41
Remn-Min Guo, “ Two Dimensional Transient Thermal Behavior of Work Rolls,” ASME, Journal
of Manufacturing Science and Technology, 1998, Vol. 120, pp. 28-33.
Sheppard, T. and Wright, D. S., ”Structural and Temperature Variations During Rolling of
Aluminum Slabs,” Metals Technology, 1980, Vol. 7, pp. 274-281.
Tseng, A. A., “A Numerical Heat Transfer Analysis of Strip Rolling,“ ASME, Journal of Heat
Transfer, 1984, Vol. 106, PP. 512-517.
Tseng, A. A., “A Generalized Finite Difference Scheme for Convection Dominated Metal Forming
Problems,” International Journal for Numerical Methods in Engineering, 1984, Vol.20, pp. 1885-
1900.
Tseng, A. A., Tong, S. X., Maslen, S. H. and Mills, J. J., “ Thermal Behavior of Aluminum Rolling,”
ASME, Journal of Heat Transfer, 1990, Vol. 112, pp. 301-308.
Zienkiewicz, O. C., Onate, E. and Heinrich, J. C., “A General Formulation for Coupled Thermal
Flow of Metals Using Finite Elements,” International Journal for Numerical Method in
Engineering, 1981, Vol. 17, pp. 1497-1514.
NOMENCLATURE
Symbol Description Unit
Bi Biot Number. -
c Specific Heat. KJ/kg.oC
F Tangential Force. N
H Heat Transfer Coefficient. W/m2 .oC.
h Half Thickness of Strip. m
k Thermal Conductivity. W/m.oC
P Pressure. N/m2
Pe Peclet Number. -
Q Heat Generation. kW
q Heat Generation Rate, Heat Friction Rate. kW/m3 or
Journal of Engineering Volume 13 September2006 Number 3
8311
kW/m2
R Roll Radius. m
r, θ Cylindrical Coordinate. -
Re Reynolds Number. -
T Temperature. oC
t Thickness. m
u, v Horizontal and Vertical Velocity. m/s
V Velocity. m/s
W The Width of the Strip. m
x,y Cartesian Coordinate. m
Abbreviations
CFDM Conventional Finite Difference Method. -
Coef. Coefficient. -
Eq. Equation. -
Fig. Figure. -
GFDM Generalized Finite Difference Method. -
Ref. Reference. -
Temp. Temperature. oC
Tran. Transfer. -
Greek Symbols
α Thermal Diffusivity. m2/s
ρ Density. kg/m3
ω Roll Angular Velocity. rad/s
δ Heat Penetration Depth. m
Ф Bite Angle. Degree
σ Plain Strain Yield Stress. N/m2
ε Local Strain Rate. 1/s
β Angle Specified the Direction. Degree
Journal of Engineering Volume 13 September2006 Number 3
1792
MATHEMATICAL SIMULATION OF FLOW THROUGH
HOLLOW FIBRE MEMBRANE UNDER CONSTANT
HYDRAULIC CONDUCTIVITY
Riyadh Zuhair Al-Zubaidy
Asst. Prof.
Department of Water Resources Engineering, University of Baghdad
ABSTRACT
Distilled water flow through a virgin hollow fibre membrane, HFM is considered as steady
nonuniform since the fibre’s wall hydraulic conductivity coefficient is kept constant along the fibre
and unchanged during the operation. Under these conditions, two well known laws were used to
mathematically simulate the hydraulic flow through the HFM. These two laws are: the Darcy’s
Law, to simulate the flow thought the fibre wall, and the Hagen-Poiseuille’s Law, to simulate the
laminar flow through the fibre channel.
Laboratory measurements were carried out to provide necessary data for the calibration
and verification of the mathematical model that was developed based on the Darcy’s and
Poiseuille’s laws. A good agreement was obtained between the measured and predicted flowrate
values under the same conditions.
The developed Mathematical model can be used as a tool to investigate the hydraulic
performance of commercial HFM modules. A comparison was made between two commercially
available of HFM modules of the same material but differ in the fibre sizes; it was found that there
is a difference between its performance and the efficiency of the operation energy.
الخلاصة ةمستقرا لثبات الايصالية الهيدروليكي يكون جريانا الجريان في الأغشية الليفية المجوفة البكر عند استخدام ماء نقي فان
ثيل الجريان خلال الأغشية الليفية فين لتمو معر قانونين استعمال أمكنتتغير إثناء التشغيل. في هذه الحالة لا على امتداد الليف و بويزل لتمثيل الجريان الطباقي خلال قناة الليف. -انون دارسي لتمثيل الجريان خلال جدار الليف وقانون هايكنالمجوفة هما ق
برهنة صحةأجريت قياسات مختبريه على الجريان خلال الأغشية الليفية المجوفة لتوفير البيانات الضرورية لمعايرة و وتلك المستنبطة من ةالمختبريبين القياسات جيد التطابق كان سي وبويزل.ار دعمل النموذج الرياضي المعد اعتمادا على قانوني
. الظروفتحت نفس الرياضي النموذج
مقارنة ت يمكن استعمال النموذج الرياضي المعد كأداة لتحري الأداء الهيدروليكي للأغشية الليفية المجوفة التجارية. تمهنالك اختلاف إنووجد من نفس النوع مع اختلاف في ابعاد الليف الأغشية الليفيةمن متوفرة تجاريا الأداء الهيدروليكي لمنتجين
. اللازمة للتشغيلالطاقة كفاءة الأداء وفي كبير في
KEYWORDS : Hollow fibre membrane, Mathematical simulation, Mathematical Model
INTRODUCTION:
R. Z. Al-Zubaidy mathematical simulation of flow through hollow fibre
membrane under constant hydraulic conductivity
1793
Hollow fibre membrane, HFM, shows a number of advantages over traditional water
filtration technique which makes it attractive to potable water industry, with the high water quality
produced using the HFM, which meets the stringent potable water regulations, makes the use of
HFM to grow rapidly within the last decade.
Several different commercial HFM modules, used for microfiltration and ultrafiltration
treatment of water, are available in the market. Even if they are made of the same material, they
differ in their capacity, diameter and length of the fibre, number of fibres used, and pot length.
Many theoretical and experimental studies were carried out to evaluate the performance of
the HFM and a number of mathematical models based on different hydraulic relations and
simplification assumptions were developed.
The main objective of this study is to develop and verify a mathematical model to simulate
the flow through the HFM under constant hydraulic conductivity based on the combination of
Darcy’s and Hagen-Poiseuille’s Laws. The developed mathematical model with the solution
procedure applied on computer is a useful tool for engineers to examine the performance of the
HFM.
MATHEMATICAL SIMULATION OF THE FLOW THROUGH HFM
The mathematical simulation of steady flow through the HFM under constant hydraulic
conductivity was developed based on two basic formulas governing the flow through the membrane
wall and the fibre channel as described in the following sections.
Flow through HFM Wall
The flow through the HFM wall is a radial flow, Fig. (1). An expression for the Darcy’s law
for redial flow through the wall of a fibre segment of ∆S length can be derived from the basic
Darcy’s formula by transforming its cartesian coordinate system into polar coordinate system, that
is:
q =
)ln(
2
n
t
r
r
TMBKS
(1)
Where
q = volumetric flowrate, L3/T,
K= hydraulic conductivity of the porous media, L/T,
TMP = transmembrane pressure head, L,
K= hydraulic conductivity of the porous media, L/T,
rt= fibre outer radius, L, and
rn= fibre inner radius, L.
The hydraulic conductivity, K, is a measure of the ability of water to flow through a porous
medium. It depends on the fluid properties and the porous medium properties through the intrinsic
permeability.
Journal of Engineering Volume 13 September2006 Number 3
1794
rn rt
∆S
Fig. (1). Schematic diagrams of longitudinal cross sections along a HFM.
The Head Losses through HFM Channel The head loss along the HFM channel segment can be estimated based on Poiseulle’s Law
for laminar flow. Poiseulle’s Law in term of pressure head along a HFM channel segment of ∆S
length may be written as:
4
8
n
l
rg
Sqh
(2)
In which:
hl = head loss, L,
g = gravitational acceleration, L/T2,
L = circular channel segment length, L,
μ= water viscosity, M/ (L.T), and
ρ = water density, M/L3.
APPLICATION OF DARCY’S AND POISEULLE’S LAWS TO THE FLOW OF HFM
Fig. (2) shows a schematic diagram of a single HFM in an actual fibre module. The pot
length, Lpot, is required to seal the fibres by using a special sealant of molding thermoplastic
material so that all the flowrate of the module will be thought the fibre channels outlets only.
The fibre has two outlets at both ends that mean that flow of the fibre membrane module is
symmetrical. Therefore, the flowrate calculations will be carried out on one half of the fibre and is
doubled for actual fibre flowrate.
Fig.(2). A schematic diagram of a HFM in an actual module.
By dividing the fibre membrane length under consideration into n equal segments of a
length equal to ∆S, then Eq. (1) is used to calculate the flowrate through segment as:
L Lpot Lpot
3 Segment number = 2 4 n-1 n 1
∆S
Line of symmetry Out flow Out flow
Inflow
R. Z. Al-Zubaidy mathematical simulation of flow through hollow fibre
membrane under constant hydraulic conductivity
1795
)ln(
2 11
n
t
r
r
TMBKSq (3)
By assuming the fibre segments, ∆S, is short enough so that variation of transmembrane
pressure, TMP, between the two ends of each segment is considered small and is neglected. The
TMP throughout 1st segment may be written as:
potap hlhTMP 1 (4)
In which:
hap = applied pressure head, L, and
hlpot= head loss through the pot length, L.
The head loss throughout the fibre channel along the pot length may be calculated by
applying Poiseulle’s Law, Eq. (2), that is:
4
21 ).........(8
n
potn
potrg
Lqqqhl
(5)
Where Lpot is the pot length, L.
Now, The expression for flowrate through 1st segment, Eq. (3), may be written as:
)ln(
))........(8
(24
21
1
n
t
n
potn
ap
r
r
rg
LqqqhKS
q
(6)
Rearranging and rewriting
0).......()1( 132212 CqqqCqC n (7)
In which
)ln(
21
n
t
ap
r
r
hKSC (8)
and
)ln(
164
2
n
t
n
pot
r
rrg
LKSC
(9)
In general, Eq. (7) may be written as:
Journal of Engineering Volume 13 September2006 Number 3
1796
0)1( 1
2
212
CqCqCn
i
i (10)
In which i is the segment number.
A similar expression can be obtained for the 2nd
segment. The TMP along the segment can
be written as:
11 hlhlhTMP potap (12)
In which hl1 is the head loss along the 1st segment, which may be obtained by using Poiseulle’s
Law, Eq. (2), that is:
4
211
).........5.0(8
n
n
rg
Sqqqhl
(13)
When calculating the head loss thought 1st segment, the flowrate thought is not fully
developed it varies from o to q1. Then it was assumed that the flowrate through the walls of this
segment varies linearly along the segment and the average was taken to calculate the head loss.
Then expression for flowrate through segment 1 may be written as:
)ln(
))......5.0(8)........(8
(24
321
4
21
2
n
t
n
n
n
potn
ap
r
r
rg
Sqqqq
rg
LqqqhKS
q
(14)
By defining
)ln(
164
2
3
n
t
r
rrg
KSC
(15)
and arranging and rewriting
0).......)(()1()5.0( 1433223212 CqqqCCqCCqC n (16)
or
0)()13()5.0( 1
3
322212
CqCCqCCqCn
j
j (17)
A similar equation may be obtained for the remaining segments, which may be written in general
form as:
0)))1(()13)1(()))5.0)1((( 1
1
32232
1
1
CqCiCqCiCqCjCn
ij
jij
i
j
(18)
R. Z. Al-Zubaidy mathematical simulation of flow through hollow fibre
membrane under constant hydraulic conductivity
1797
By applying Eq. (10) to the 1st segment and Eq. (18) to the 2
nd segment through the n
th
segment, the resultant is a system of linear equations with n unknowns represents the flowrate
throughout each segment, which may be written as shown in Table 1.
Table 1. The resulting system of linear equation.
Equation
no. q1 q2 q3 q4
qn R.H.S
1 C2+1 C2 C2 C2 C2 C1
2 C2+0.5 C3 C2+C3+1 C2+C3 C2+C3 C2+C3 C1
3 C2+0.5 C3 C2+1.5C3 C2+2C3+1 C2+2C3 C2+2C3 C1
4 C2+0.5 C3 C2+1.5C3 C2+2.5C3 C2+3C3+1 C2+3C3 C1
n C2+0.5 C3 C2+1.5C3 C2+2.5C3 C2+3.5C3 C2+(n-1)C3+1 C1
The above system of n simultaneous equations can be solved for the flowrate using any
method for solving a system of linear equations. Having obtaining the values of the flowrate of each
segment, the transmembrane pressure can be calculated for each segment.
MATHEMATICAL MODEL VERIFICATION
The developed mathematical model was calibrated and verified by using gathered
laboratory experimental data to check the performance of the model and the validity of the
assumptions made.
Laboratory experiments were carried out on polypropylene, PP, HFM with inner and outer
diameters of 0.39mm and 0.65mm, respectively. The fibres were divided into four sets each set
consist of ten fibres. The flowrate, under a constant head of 2m, of each set was measured with its
initial length, and then the flowrate is measured each time after reducing the length as shown in
Table (2).
Table (2). Measured flowrate values.
Set no. Length
(cm)
Measured flowrate
(ml/min) Set no.
Length
(cm)
Measured flowrate
(ml/min)
1
50 18.1
3
50 18.6
40 17.02 40 16.82
30 14.84 30 14.84
20 11.2 20 11.89
10 5.9 10 6.48
2
45 18.1
4
45 18.14
30 15.23 30 14.41
15 8 15 7.84
The first step toward the mathematical model verification is the calibration of the hydraulic
conductivity coefficient. The value of this coefficient was calibrated using the data of the first set
with initial length of 50cm only. The conductivity coefficient is adjusted until the predicted
Journal of Engineering Volume 13 September2006 Number 3
1798
flowrate value matches the experiment value. The conductivity coefficient value was found to be
4.64*10-7
cm/s.
The Mathematical model was used then to generate the flowrate values of the fibre sets by
just changing the fibres length. Fig. (3) shows the measured and mathematical model predicted
flowrate values. A segment length of 1cm was adopted in all calculations of the study. A good
agreement between the measured and predicted flowrates values can be noticed with a correlation
coefficient of 0.996.
10 20 30 40 50
Fibre length (cm)
0
5
10
15
20
Flo
wra
te (
ml/
min
)
Figure. (3). Comparison between measured and predicted flowrate values.
Measured flowrate
Predicted flowrate
Fig. 3. Comparison between measured and predicted flowrate values.
APPLICATION OF THE MATHEMATICAL MODEL
The developed mathematical model being verified can be used to study the hydraulic
performance of HFM rather than carrying out time consuming laboratory tests.
The mathematical model was applied to investigate the hydraulic performance of two types
of virgin HFM modules.
First Type of HFM Module
Specifications of the first type HFM module are listed in Table 3.
Table 3. Specification of the first type HFM module.
Item Value
Fibre inner diameter 0.25mm
Fibre outer diameter 0.55mm
Number of fibres per module 20 000
Effective fibre length 97cm
Pot length 10cm
Total effective area 33.5m2
Normal Module operation flowrate 120-240 lmh/bar
In the factory, the measured flowrate of the fibres module with RO permeate is 2,000
lmh/bar at 20°C. This given permeability can be reached by the mathematical model under a
hydraulic conductivity coefficient of 2.32x10-5
m/s.
R. Z. Al-Zubaidy mathematical simulation of flow through hollow fibre
membrane under constant hydraulic conductivity
1799
The flowrate variation as a percentage of the total fibre flowrate a long a single fibre length
is shown in Fig. 4. As it may be seen that more that 99.3% of the flowrate is just from the first
10cm of the fibre length.
10 20 30 40 50
Segment no.
0
10
20
30
40
% o
f to
tal
flow
rate
Fig. (4). Flowrate through each computational segment a long the fibre length as a percentage of the total fibre flowrate
Distance (cm)
Fig. 4. The flowrate variation as a percentage of total flowrate along a single fibre length.
Fig. 5 shows the TMP variation as a percentage of the total applied pressure head along
the single fibre length. It is clear that most of the applied head will be used to derive water from the
first 10cm.
10 20 30 40 50
Segment no.
0
10
20
30
40
50
60
70
80
90
100
TM
P a
s %
of
the
ap
pli
ed P
ress
ure
hea
d
Fig. (5). TMP of each computational segment a long the fibre length as a percentage of the applied pressure head.
Distance (cm)
Fig 5. The TMP variation as a percentage of the total applied head along a single fibre
length.
The flowrate of a single fibre is reduced when placed in actual fibre module because of the
headloss through the module pot. The flowrate of the single virgin fibres will be reduced from 2000
lmh/bar down to 336.4 lmh/bar when placed in a full module.
Journal of Engineering Volume 13 September2006 Number 3
1800
The TMP variation as a percentage of the total applied pressure head and flowrate variation
as a percentage of the total flowrate along the fibre length are independent of the applied pressure
head. Thus, the flowrate variation along the fibre as a percentage of total flowrate will be the same
as in Fig. 4.
Fig. 6. Shows the TMP variation along the fibre with a pot of a 10cm length. A great
reduction in the TMP may be noticed when comparing the TMP variation of Fig. 5, of a single
fibre, with that of Fig. 6, a fibre in an actual module. 83.2% of the total applied energy will be lost
through the pot length.
0 10 20 30 40 50
Distance (cm)
0
5
10
15
20
TM
P a
s %
of
the
ap
pli
ed P
ress
ure
hea
d
Fig. (6). TMP of each computational segment a long the fibre length as a percentage of the applied pressure head.
FIG 6. The TMP variation as a percentage of the total applied head along the fibre length.
Second Type of HFM Module
Specifications of the second type of HFM module are listed in Table 4.
Table 4. Table 3. Specification of the second type HFMe module.
Item Value
Fibre inner diameter 0.8mm
Fibre outer diameter 1.2mm
Effective fibre length 144.75cm
Pot length 4cm
Total effective area 35m2
Module Max operation flowrate 357 lmh/bar
The hydraulic conductivity coefficient was found to be 2.57*10-7
cm/s at which max
operation permeability was reached.
The flowrate variation a long the fibre length as a percentage of the total fibre flowrate is
shown in Fig. 7. As it may be seen that the ratio between the flowrate at the module outlet and that
at its middle is about 1.1%.
R. Z. Al-Zubaidy mathematical simulation of flow through hollow fibre
membrane under constant hydraulic conductivity
1801
10 20 30 40 50 60 70 80
Segment no.
1.0
1.1
1.2
1.3
1.4
1.5
% o
f to
tal
flow
rate
Fig. (7). Flowrate through each computational segment a long the fibre length as a percentage of the total fibre flowrate
Distance (cm)
Fig. 7. The flowrate variation along the fibre length as a percentage of total flowrate.
Fig. 8 shows the TMP variation along the fibre length as a percentage of the total applied
pressure head. Due to large fibre diameter and the short length of the pot the head losses through it
is very small. The difference between the maximum and the minimum TMP along the fibre is less
than 10%.
10 20 30 40 50 60 70 80
Segment no.
80
90
100
TM
P a
s %
of
the
ap
pli
ed P
ress
ure
hea
d
Fig. (8). TMP of each computational segment a long the fibre length as a percentage of the applied pressure head.
Distance (cm)
FIG 6. The TMP variation along the fibre length as a percentage of the total applied head.
CONCLUSIONS - The flow though the HFM under the condition of constant hydraulic conductivity could be
simulated mathematically by applying the Darcy’s Law and Poiseulle’s Laws. A Good agreement
was found between the laboratory and predicted data under the same conditions.
- A great difference in the performance of two the commercial HFM was noticed.
- The TMP variation as a percentage of the total applied pressure head and flowrate variation as a
percentage of the total flowrate along the HFM length are independent of the applied pressure head
- The head losses through the pot length could be so high and consumes 80% of the total applied
pressure head.
Journal of Engineering Volume 13 September2006 Number 3
1802
RECOMMENDATIONS
The following recommendations were suggested to study:
- The variation of the hydraulic conductivity along the fibre length under normal operation
conditions.
- The mathematical simulation of the hydraulic flow through the hollow fibre membrane under
variable hydraulic conductivity.
- Optimizing the HFM module design.
REFERENCE
- Baker, R. W., Membrane Technology and Applications, 2nd Edition, John Wiley and Sons Ltd,
England, 2004.
- Technical data provided by the Technical Director for Memcor Ltd, United Kingdom, part of US
Filter company, 2005.
- W. S. Winston HO and Kamaleh K. SirkAr, “Membrane Hand book”, 1992, Van Nostrand
Reinhold, New York.
- X-Flow B.V., The XIGATM Concept and Operation Manual, 2005, at http://www.x-
flow.com/import/assetmanager/1/3161/CAPF-XIGA-0143.pdf.
LIST OF SYMBOLS
∆S= length of the fibre segment, L.
μ= water viscosity, L.T.
HFM= hollow fibre membrane.
hap = applied head, L.
hl = head loss, L.
hlpot= pressure head loss through the pot length, L.
K= hydraulic conductivity of the porous media, L/T.
Lpot = length HFM module pot, L
q = volumetric flowrate, L3/T.
rt= fibre outer radius, L, and
rn= fibre inner radius, L.
TMP = transmembrane pressure, L.
Journal of Engineering Volume 13 September2006 Number 3
1701
EFFECT OF TRANSVERSE BASE RESTRAINT ON THE
CRACKING BEHAVIOR OF MASSIVE CONCRETE
Dr. Adnan F. Ali Dr. Omar Al-Farouq Al-Damluji Ala’a H. A. Al-Zuhairi
Ass. Professor
Baghdad University Ass. Professor
Baghdad University Lecturer
Baghdad University
ABSTRACT
The effect of considering the third dimension in mass concrete members on its cracking
behavior is investigated in this study. The investigation includes thermal and structural analyses of
mass concrete structures. From thermal analysis, the actual temperature distribution throughout the
mass concrete body was obtained due to the generation of heat as a result of cement hydration in
addition to the ambient circumstances. This was performed via solving the differential equations of
heat conduction and convection using the finite element method.
The finite element method was also implemented in the structural analysis adopting the
concept of initial strain problem. Drying shrinkage volume changes were calculated using the
procedure suggested by ACI Committee 209 and inverted to equivalent temperature differences to be
added algebraically to the temperature differences obtained from thermal analysis.
Willam-Warnke model with five strength parameters is used in modeling of concrete material
in which cracking and crushing behavior of concrete can be included. The ANSYS program was
employed in a modified manner to perform the above analyses.
A thick concrete slab of 1.5m in thickness and 10m in length was analyzed for different widths
2, 4, 8, and 10m to produce different aspect ratios (B/L) of 0.2, 0.4, 0.8, and 1.0 respectively. The
results of the analyses show an increase in cracking tendency of mass concrete member as the aspect
ratio of the same member is increased due to the effect of transverse base restraint. Accordingly, such
effect cannot be ignored in the analysis of base restrained mass concrete structures subjected to
temperature and drying shrinkage volume changes.
الخلاصةفييه اييلب ب، تييإج اييت رييينبت اتيينخ ييب اييلثان فييل ب، ييخ ب،ثة،ييإ فييه رييةت ب،فنعييةتش ب،اا اييش ت يين ب ا ييةن يي ا يي ة
ت ا ييه ،ييخنيةت ب،تيينبن وااريي ب ب،خنبعييش رييينبت ب،ات اييي ب،تيينبنخ وبشت ييةخه ، ت ييكت ب،فنعييةتش ب،اا اييش واييات ب،ت ييوي يي ب،او ايي ب،فيي ي يعييت ب،فنعييةتش ب،اا اييش ييب فيي ي ب،ات اييي ب،تيينبنخ ، عييل،ش تاايييش ، تيينبن ب، او،ييخ ييينبت ر ةاييش ب،عيي تت ةشرييةفش ر،يي ب، يينو
ب،يواش ب، تاطش والب اينى ب ف ي تي ب، ةخ،ااب ب،افةر اااب ، او اي وب،ت ي ب،تنبنخ ةعافخبت طنا ش ب، تة ن ب، تخخ ارة فه ب،ات اي بلأت ةخه ول،ك ا ته فاين عيل،ش بلأتف يةي ب ايخبخه ايل،ك ايات عافخ تا ة رب طنا ش ب، تة ن ب، تخخ
902بتاعةب ب،ا ةت ب،تي اش ب، نبف ش ،يفة ب،فنعةتش ة ا يةخ ي ب،طنا يش ب، انتيش يب ع يي يخ ب،فنعيةتش بلأ ناايه ،يتيش ااةفخ ة ب اغان فه خنيةت ب،تنبن ،ارة ي ناة ر، ب،اغانبت خنيةت ب،تنبن ب،ايه ايت ب،ت يوي ا ية يب وب،اه اات اتوا ة ر، ة
ب،ات اي ب،تنبنخ ةو يييش ،ا ثايييي يييةخ ب،فنعيييةتش رل اارييي ب ايييلب ب،ت يييول ا ييين تف عيييش ييية لبتونبتيييك -، يييخ ايييت ا تيييه ت يييول واييي ت
ش شتية ب،ات ا اب ب، ةن ،ا ة ه خ،و طنا ش ANSYSت بعافخبت نتة ج الب وا تب،فنعةتش فه تة،اه ب،ا ق وب،ا
A.F. Ali Effect of Transverse Base Restraint on O. Al-Farouq Al-Damluji the Cracking Behavior of Massive Concrete
A. H. A. Al-Zuhairi
2071
ت ،ات اييق تعييب يةت اييش نم طييوي 50و 8ج 4ج 9ت و ييل نبم فا فييش 50ت وطييوي 5.1 عيي ك طييش فنعييةتاشاييت ات اييي ةت ب،فنعييةتش ب،اا اييش اييةخ ب،تعييب يي ب،انااييب يينت تاييةخج ب،ات اييي اييةخ فييه عة اييش ا يي ق ريي 0ج5و 8ج0ج 4ج0ج 9ج0 فا فييش
ب،يةت ايش عي ب ايلثان ا اايخ ب، ة يخ ب، نريه و ايمج فيلب ثيي ايلب ب،ايلثان ا ايب را ة،يم فيه ت يكت ب،فنعيةتش ب،اا ايش ب،ايه اا ينم ،اغانبت فه خنيةت ب،تنبن و ا ص ب،يفة
KEYWORDS: Mass Concrete, Temperature Difference, Drying Volume Change, Base Restraint,
Concrete Cracking.
INTRODUTION
Mass concrete is an expression usually used for any concrete structure with dimensions large
enough to cause structural problems during and after the construction period. These problems are
mainly the occurrence of cracking due to temperature variations and shrinkage volume changes. Like
any solid material, concrete is affected by increase and decrease of temperature. The effect appears as
a thermal strain that occurs within the concrete structure when it is prevented or restricted from
motion, i.e., restrained. The second category of volume change is the drying shrinkage, which is
related to the drying and shrinking of the cement gel.
ACI 207 Committee (ACI Committee 1995) suggested the following equations to be used to
calculate the degree of restraint for rigid continuous base restraint.
5.2HLfor10HL1HLK
5.2HLfor1HL2HLK
HhR
HhR
(1)
where
L/H = length to height ratio, and
h = the height at which the degree of restraint is calculated.
As can be noticed, ACI 207 Committee neglects the effect of the restraint in the transverse
direction and hence, eq. (1) can be applied to the concrete walls only. Therefore, it is the objective of
the present study to investigate the effect of transverse base restraint, i.e., effects of 3rd
dimension on
the behavior of massive concrete and therefore cracking tendency and cracking prevention in such
structures.
THERMAL ANALYSIS
Based on Fourier’s Law for heat transfer, the heat conduction equation can be expressed as
follows (Holman, 1981):
t
Tcq
z
Tk
y
Tk
x
Tk p2
2
z2
2
y2
2
x
(2)
where,
kx, ky and kz = heat conductivity of the material in x, y and z-direction respectively,
T = the difference between absolute and reference temperatures,
q = heat generation per unit volume,
cp = specific heat, and
= density.
On the other hand, heat may transfer by convection according to the following Newton
formula (Holman, 1981):
Journal of Engineering Volume 13 September2006 Number 3
1703
dATThqA
cv (3)
where,
cvh = convection heat transfer coefficient (film coefficient).
T = bulk fluid temperature.
T = temperature.
Both of the above equations may be descritized using Raylieh-Ritz variation process to derive
an expression employing the finite difference method to overcome the time-rate nature of the problem,
that is,
*t
*e FK (4)
where,
ttt
*
e*e
t
CFF
t
CKK
Eq. (4) is used in the finite element method to predict the temperature distribution within the mass
concrete body invoking the described initial and boundary temperatures as follows:
1. Initial temperature = concrete placement temperature = 20 oC.
2. Bulk ambient temperature T which is specified for Baghdad climate according to Kammouna,
2001, from:
T = 29.815 – 15.291*cos(0.893t) (5)
where, t is time in days.
SHRINKAGE STRAIN CALCULATION Following the procedure recommended by (ACI Committee 209, 1992) and taking the ambient
circumstances and concrete mixing and placing conditions, the drying shrinkage strains may be
calculated as a function of time after curing period for concrete which is assumed to be seven days.
Quoting the concept of evaporable moisture content that was adopted by Carlson, 1937, the
distribution of drying shrinkage strains may be assessed within the body of mass concrete member.
Table (1) shows such distribution in which the values of drying shrinkage strains seem very low. Such
observation may be related to the non-convenience of eq. (6) that was suggested by ACI 209
Committee (ACI Committee, 1992) and used to estimate shrinkage strain-time relation.
ushtsh
t35
t
(6)
where,
(sh)t = shrinkage strain at any time t (in days), and
(sh)u = ultimate shrinkage strain = 780*10-6
.
As can be seen from eq. (6) 50% only of the ultimate shrinkage strain occurs at 35 days after curing.
However, one can conclude from the trend of the drying shrinkage strains as they are decreased with
the increase in the width of the slab as the major cause of cracking in large mass concrete members is
the temperature variation rather than shrinkage volume changes.
A.F. Ali Effect of Transverse Base Restraint on O. Al-Farouq Al-Damluji the Cracking Behavior of Massive Concrete
A. H. A. Al-Zuhairi
2071
ADOPTED COSTITUTIVE RELATIONSHIP FOR CONCRETE The concrete was modeled using Willam and Wranke model (Willam and Wranke, 1974)
which predicts failure of brittle materials in which the cracking and crushing modes should be
accounted for. The criterion for failure of concrete due to a multiaxial stress state can be expressed in
the form:
0Sf
F
c
(7)
where,
F = a function of the principal stress state (xp, yp, zp),
S = failure surface expressed in terms of principal stresses and five input strength parameters
as follows:
ft = ultimate uniaxial tensile strength in MPa
fc = ultimate uniaxial compressive crushing strength in MPa,
fcb = ultimate biaxial compressive strength in Mpa,
h = ambient hydrostatic stress state in MPa,
f1 = ambient hydrostatic stress state of biaxial superimposed on hydrostatic stress state in Mpa,
and
f2 = ambient hydrostatic stress state of uniaxial superimposed on hydrostatic stress state in
Mpa.
For simplicity, Willam and Warnke, 1974, suggested the following equations to calculate three
of strength parameters in terms of fc in case of ch f3 . Thus, the failure surface S can be
specified with a minimum two constants, ft and fc
.
c2
c1
ccb
f725.1f
f45.1f
f2.1f
(8)
Failure of concrete is categorized into four domains. In each domain, independent functions
were specified to describe the function F and the failure surface S. The failure surface S can be seen in
Fig. (1).
Compression-Compression-Compression Domain(0 1 2 3)
In this case, F takes the form:
21
213
232
2211
15
1FF (9)
and the failure surface S is defined as:
212
221
22
2
1
212
122
12
22122
12
221
r2rcosrr4
rr4r5cosrr4rr2rcosrrr2SS
(10)
where,
Journal of Engineering Volume 13 September2006 Number 3
1705
21
213
232
221
321
2
2cos
in which = angle of similarity, and
c
h
22102
22101
f
bbbr
aaar
The undetermined coefficients a0, a1, a2, b0, b1, and b2 are discussed below.
When = 0o, S1 in eq. (10) is equal to r1 while if = 60
o, S1 is equal to r2. Therefore, the
function r1 represents the failure surface of all stress states with = 0o.
The function r1 is determined by adjusting a0, a1, and a2 such that ft , fcb and f1 all lie on the
failure surface. Mathematically:
2
1
0
211
2cbcb
2tt
1ah32
ah1
c
1
cb321
c
1
32t1
c
1
a
a
a
1
1
1
f,f
F
f,0f
F
0,ff
F
(11)
in which .f3
f2
fand,
f3
f,
f3
f
c
1
c
ah
1
c
cbcb
c
tt
The proper values for the coefficient a0, a1, and a2 can be determined through the solution of the
simultaneous equations given in eq. (11).
The function r2 is calculated by adjusting b0, b1, and b2 to satisfy the conditions:
2
1
0
200
2222
ah3
ah21
c
1
c321
c
1
b
b
b
1
1
9
1
3
11
0
f,f
F
f,0f
F
(12)
where,
2 is defined by: .f3
f
f c
2
c
ah
2
and 0 is the positive root of the equation:
20201002 aaar (13)
in which a0, a1, and a2 are evaluated by eq. (11).
A.F. Ali Effect of Transverse Base Restraint on O. Al-Farouq Al-Damluji the Cracking Behavior of Massive Concrete
A. H. A. Al-Zuhairi
2071
Since the failure surface must remain convex, the ratio r1/r2 is restricted to the range
(0.5<r1/r2<1.25), although the upper bound is not considered to restriction since (r1/r2 < 1.0) for most
materials. Also, the coefficients a0, a1, a2, b0, b1, and b2 must satisfy the conditions:
a0 > 0, a1 0, a2 0
b0 >0, b1 0, b2 0
Therefore, the failure surface is closed and predicts failure under high hydrostatic pressure (< 2).
This closure of the failure surface has not been verified experimentally and it has been suggested that
Von Mises type cylinder is a more valid failure surface for large compressive h-values.
Consequently, it is recommended that values of f1 and f2 are selected at a hydrostatic stress level in the
vicinity of or above the expected maximum hydrostatic stress encountered in the structure.
Eq. (9) describes the condition that the failure surface has an apex at = 0. A profile of r1 and
r2 as a function of is shown in Fig. (2).The lower curve represents all stress state such that = 0o
while the upper curve represents stress state such that = 60o. If the failure criterion is satisfied, the
material is assumed to crush. Tension-Compression-Compression Domain (1 0 2 3)
In this regime, F takes the form:
21
23
22
2322
15
1FF (14)
and S is defined as:
212
221
22
2
1
2121
221
22212
21
222
t
12
p2pcospp4
pp4p5cospp4pp2pcosppp2
f1SS
(15)
where cos is already defined above, and
32
22102
22101
3
1
bbbp
aaap
The coefficients a0, a1, a2, b0, b1, and b2 are defined by eq. (11) and eq. (12).
If the failure criterion is satisfied, cracking occurs in the plane perpendicular to the principal
stress 1.
Tension- Tension -Compression Domain (1 2 0 3)
Here the function F takes the form:
2,1i;FF i3 (16)
and the failure surface S is defined as:
2,1i;,0,S
1f
fSS
3i2
3
c
t3
(17)
Journal of Engineering Volume 13 September2006 Number 3
1707
If the failure criterion for both i = 1, 2 is satisfied, cracking occur in the planes perpendicular
to principal stresses 1, 2. If the failure criterion is satisfied only for i = 1, cracking occurs only in the
plane perpendicular to principal stress 1.
Tension- Tension - Tension Domain (1 2 0 3)
In this regime, F takes the form:
3,2,1i;FF i4 (18)
and S is defined as:
c
t4
f
fSS
(19)
If the failure criterion is satisfied in directions1, 2, and 3, cracking occurs in the planes
perpendicular to principal stresses 1, 2, and 3, otherwise, cracking occurs in plane or plane
perpendicular to the directions of principal stresses where the failure criterion is satisfied.
IMPLEMENTATION OF THE FINITE ELEMENT METHOD
According to Fung, 1965, the effect of temperature changes on an elastic body subjected to
external forces may be determined using one of the followings:
1. Solution of the discretized form of the coupled thermo-elastic equation in which the effect of both
temperature and displacement on each other may be determined, i.e. the displacement due to unit
temperature change and vice versa. However, this procedure is not usually used especially in problems
where the temperature changes are not high enough like in mass concrete problem.
2. When the simplifying assumptions mentioned in (1) above are introduced, the theory is referred to
as an uncoupled, quasi-static theory; it degenerates into heat conduction and thermoelasticity as two
separate problems. Experience shows that the change of temperature of an elastic body due to
adiabatic straining is, in general, very small. If this interaction between strain and temperature is
ignored, then the only effects of elasticity on the temperature distribution are effects of change in
dimensions of the body under investigation. The change in dimension of a body is of the order of
product of the linear dimension of the body L, the temperature rise T, and the coefficient of thermal
expansion . If L = 1m and T = 100 oC, = 10*10
-6 per
oC, the change in dimension is 10
-3m, which
is negligible in problems of heat conduction.
The equivalent temperature changes to the estimated drying shrinkage strains may be
calculated using the following simple relation:
c
shDST
(20)
where,
TDS = drying shrinkage equivalent temperature change,
sh = shrinkage strain, and
c = coefficient of thermal expansion of concrete.
Then the equivalent temperature changes to drying shrinkage may be added algebraically to
the temperature changes resulting from thermal analysis. The effect of this sum of temperatures,
which appears as thermal stress and strain, may be detected using the second method described in (2)
above. This means that the problem is treated as “an initial stress or strain” problem. The term “initial
stress” signifies a stress present before deformations are allowed. Effectively, it is a residual stress to
A.F. Ali Effect of Transverse Base Restraint on O. Al-Farouq Al-Damluji the Cracking Behavior of Massive Concrete
A. H. A. Al-Zuhairi
2071
be superposed on stress caused by deformation. The effect of temperature changes can be placed as
initial strain o, or initial stress and strain o and o. Both are viewed as alternative ways to express the
same thing (Cook, 1989).
In a linear elastic material, the stress-strain relation is (Cook, 1989):
oE
(21)
where,
o = initial strain = cT
or,
oE (22)
The strain energy Uo is defined as (Cook, 1989):
dv2
Uv
o
(23)
Substituting eq. (22) into eq. (23):
dv2
EU
2o
vo
or,
dv22
EU 2
oo2
vo (24)
The third term in the parenthesis in eq. (24) can be omitted since it is independent of nodal
displacements. Then its derivative is equal to zero. Thus,
dv22
EU o
2
vo (25)
Writing dv22
Eu o
2o = is the energy per unit volume. Hence, for a state of multiaxial stresses:
oo DD2
1u (26)
where
[D] = the constitutive relations matrix for concrete and is defined for linear-elastic material as
follows (Cook, 1989):
Journal of Engineering Volume 13 September2006 Number 3
1709
2
2100000
02
210000
002
21000
0001
0001
0001
211
ED
(27)
The potential of external load may be expressed as:
uex (28)
where,
u = the displacement field vector, and
= the load vector.
Furthermore, the potential of body forces is given by:
Fubf (29)
where
F = the body force vector, which is any force distributed over the entire volume of the body
like the self-weight.
The total potential energy per unit volume can be written in the form (Cook, 1989):
FuuDD2
1U oop0
(30)
The total potential within the element is:
v
pp dvoe
(31)
or,
vsv v
op dvFudsudvDdvD2
1e
(32)
Since,
eNu (33)
where,
[N] = shape function matrix, and
e = nodal displacement vector.
A.F. Ali Effect of Transverse Base Restraint on O. Al-Farouq Al-Damluji the Cracking Behavior of Massive Concrete
A. H. A. Al-Zuhairi
2027
Also,
eB (34)
where,
[B] = strain-nodal displacement matrix.
Hence,
dvFNedsNedvDBedveDBe2
1
vs
To
vv
Tpe
(35)
which after simplification and introducing effects of externally applied nodal forces becomes:
FeReeKe2
1ep (36)
where, F = externally applied nodal forces vector.
Applying the minimization of the total potential yields:
0
e
p
or,
RFeK (37)
where, dsNdvFNdvDBRs
T
v
T
vo
T
Eq. (37) will be used in the analysis of the mass concrete due to effects of temperature and
drying shrinkage volume changes.
COMPUTER IMPLEMENTATION
Besides the “Graphical User Interface (GUI)” that is commonly used in software packages,
ANSYS program proposes a programming language similar to some extent to the conventional
FORTRAN language. The proposed language is referred as APDL (ANSYS Parametric Design
Language).
A modified ANSYS program is adopted in this study. This consists of a main program and
four subprograms. The main program contains the principal steps of analysis and required calls for
subprograms. Each of these subprograms is responsible of some limit tasks like:
- Performing the thermal analysis,
- Storing temperature values in a pre-dimensioned array,
- Calculating drying shrinkage strains throughout the concrete body, and
- Conducting the nonlinear structural analysis by considering the effect of concrete aging via
updating concrete strength parameters (ft , fc
, Ec) after deleting the thermal finite element mesh
and constructing a new structural one.
Two Types of elements are used in this program:
Journal of Engineering Volume 13 September2006 Number 3
1711
Thermal Solid 70: This element is an eight-noded brick element with one degree of freedom,
temperature, at each node. The element is applicable to a three-dimensional, steady state or
transient thermal analysis.
Structural Solid 65:This element is used for the three-dimensional modeling of the concrete with
or without reinforcing bars. The element is defined by eight nodes having three degrees of
freedom at each node: translations in the global x, y, and z directions. It is capable of considering
cracking in tension and crushing in compression.
PROBLEM DESCRIPTION AND RESULTS
To investigate the effect of the transverse base restraint on the cracking behavior of mass
concrete member due to temperature variation and drying shrinkage volume changes, a nonlinear
finite element analysis was applied to base restrained thick concrete slab. Four different aspect ratios
(width/length) were considered for the case of a slab with fixed bottom cast at the first of January
(winter concrete placement) in Baghdad. The aspect ratios were 0.2, 0.4, 0.8, and 1.0 and obtained by
fixing the length of the slab to 10 meters and varying the width as 2, 4, 8, and 10 meters. In all these
cases, the thickness of the slab was taken as 1.5 m.
Thermal and structural analyses were conducted on the slab including all the surrounding
circumstances and boundaries utilizing the finite element mesh shown in Fig. (3).
The temperature distribution in the central sections along the length and width directions at
some times after concrete placement are shown in Figs. (4) to (15). The assessed final cracking pattern
of the concrete slabs with different aspect ratios can be seen in Figs. (16) to (19).
CONCLUSIONS
The following conclusions can be drawn from the results of the analysis:
1. Value of peak temperature increases with increasing aspect ratio (B/L) of the slab. This may be
related to the increase in the magnitude of heat generated upon concrete placement due to volume
increase.
2. A small temperature drop at the 28th
day of concrete age is noticed as the ratio B/L is increased.
This is related to the effect of the volume to surface ratio (V/S), since the temperature increases with
increasing the aspect ratio (B/L).
3. The number of primary cracks (cracks that extend over the entire thickness) increase with
increasing width of the slab, i.e., the aspect ratio. This can be interpreted as a result of the
considerable increase in restraint provided by the slab base and the increase in the maximum
temperature due to the hydration process after concrete placement.
4. The full-depth cracks are concentrated at the central portion of the slab where the maximum drop
in temperature occurs.
From the above and since it was concluded previously that the major cause of cracking in the
thick slabs is temperature drop, the effect of the third dimension (width) cannot be ignored when the
response (stresses and cracking) of this type of structures is required. Unfortunately, most of the
standards like ACI-Committee neglect the effect of the third dimension and assume a uniform
temperature distribution.
Table (1) Drying shrinkage strains
d/H
(*)
Drying shrinkage strain (10-6
)
Slab width = 2.0m Slab width = 4.0m Slab width = 8.0m Slab width = 10.0m
After … days After … days After … days After … days
A.F. Ali Effect of Transverse Base Restraint on O. Al-Farouq Al-Damluji the Cracking Behavior of Massive Concrete
A. H. A. Al-Zuhairi
2021
8 14 28 90 180 8 14 28 90 180 8 14 28 90 180 8 14 28 90 180
0 0.5 3.2 7.5 18.0 24.3 0.2 1.3 3.0 7.1 9.6 0. 1 0.6 1.4 3.3 4.4 0.08 0.48 1.12 2.7 3.6
0.2 0.2 1.3 3.1 7.5 10.3 0.1 0.5 1.2 3.0 4.1 0.04 0.25 0.57 1.4 1.9 0.03 0.2 0.47 1.13 1.5
0.4 0.07 0.4 1.01 2.4 3.2 0.03 0.17 0.4 0.96 1.3 0.01 0.08 0.19 0.44 0.63 0.01 0.07 0.15 0.36 0.48
0.6 0.02 0.1 0.3 0.8 1.1 0.01 0.05 0.12 0.3 0.45 0 0.02 0.05 0.14 0.21 0 0.02 0.04 0.11 0.17
0.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
(*) d/H defines the ratio of the depth from top surface / the slab thickness H.
zp > 0 (cracking)
zp = 0 (crushing)
zp < 0 (crushing)
yp
xp
fc
f t
ft
fc
crack
ing
cracking cracking
Fig.(1): Failure surface in principal stress space with nearly biaxial stress state, after ANSYS Inc.
= 60o
= 0o
f2r2
fc
f1
fcb
ft
r1
2
1
c
0
t
Fig. (2): A profile of the failure surface, after ANSYS Inc.
4 @
2.5
m
20 @ 0.5 m 3 @ 2 m 3 @ 2 m
Co
n.
slab
S
oil
fo
un
d.
Journal of Engineering Volume 13 September2006 Number 3
1713
Notes:
- we = concrete element width (variable) = 0.25, 0.5, 1.0, or 1.5 m. - Dimensions are not to scale.
Fig. (3): Finite element mesh for a thick concrete slab problem
Note: Temperatures are in oC
Fig. (4): Temperature distribution in a slab with B/L=0.2 (3 days after placement)
Note: Temperatures are in oC
Fig. (5): Temperature distribution in a slab with B/L=0.2 (28 days after placement)
Note: Temperatures are in oC
Fig. (6): Temperature distribution in a slab with B/L=0.2 (180 days after placement)
5 @
0.3
m
3 @ 2 m 3 @ 2 m
4 @
2.5
m
5 @
0.3
m
8 @ we
m
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
2024283236404448
Along length direction
-1.0 -0.5 0.0 0.5 1.00.0
0.3
0.6
0.9
1.2
1.5
Along width direction
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
16171819202122
Along length direction
-1.0 -0.5 0.0 0.5 1.00.0
0.3
0.6
0.9
1.2
1.5
Along width direction
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
394041424344
Along length direction
-1.0 -0.5 0.0 0.5 1.00.0
0.3
0.6
0.9
1.2
1.5
Along width direction
A.F. Ali Effect of Transverse Base Restraint on O. Al-Farouq Al-Damluji the Cracking Behavior of Massive Concrete
A. H. A. Al-Zuhairi
2021
Note: Temperatures are in oC
Fig. (7): Temperature distribution in a slab with B/L=0.4 (3 days after placement)
Note: Temperatures are in oC
Fig. (8): Temperature distribution in a slab with B/L=0.4 (28 days after placement)
Note: Temperatures are in oC
Fig. (9): Temperature distribution in a slab with B/L=0.4 (180 days after placement)
Note: Temperatures are in oC
Fig. (10): Temperature distribution in a slab with B/L=0.8 (3 days after placement)
-2 -1 0 1 20.0
0.3
0.6
0.9
1.2
1.5
Along width dierction
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
20
24
28
32
36
40
44
48
Along length direction
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
16
18
20
22
24
26
28
Along length direction
-2 -1 0 1 20.0
0.3
0.6
0.9
1.2
1.5
Along width direction
-2 -1 0 1 20.0
0.3
0.6
0.9
1.2
1.5
Along width direction-5 -4 -3 -2 -1 0 1 2 3 4 5
0.0
0.3
0.6
0.9
1.2
1.5
36
37
38
39
40
41
42
43
44
Along length direction
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
20
24
28
32
36
40
44
48
Along length direction
-4 -3 -2 -1 0 1 2 3 40.0
0.3
0.6
0.9
1.2
1.5
Along width direction
Journal of Engineering Volume 13 September2006 Number 3
1715
Note: Temperatures are in oC
Fig. (11): Temperature distribution in a slab with B/L=0.8 (28 days after placement)
Note: Temperatures are in oC
Fig. (12): Temperature distribution in a slab with B/L=0.8 (180 days after placement)
Note: Temperatures are in oC
Fig. (13): Temperature distribution in a slab with B/L=1.0 (3 days after placement)
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
16
18
20
22
24
26
28
30
32
Along length direction
-4 -3 -2 -1 0 1 2 3 40.0
0.3
0.6
0.9
1.2
1.5
Along width direction
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.534
35
36
37
38
39
40
41
42
43
44
Along length direction
-4 -3 -2 -1 0 1 2 3 40.0
0.3
0.6
0.9
1.2
1.5
Along width direction
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
28
32
36
40
44
48
52
56
60
Along length and width directions
A.F. Ali Effect of Transverse Base Restraint on O. Al-Farouq Al-Damluji the Cracking Behavior of Massive Concrete
A. H. A. Al-Zuhairi
2021
Note: Temperatures are in oC
Fig. (14): Temperature distribution in a slab with B/L=1.0 (28 days after placement)
Note: Temperatures are in oC
Fig. (15): Temperature distribution in a slab with B/L=1.0 (180 days after placement)
Side view of the central section along length direction
Top view
Fig. (16): Final cracking pattern for slab cast in winter with B/L=0.2
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
16
18
20
22
24
26
28
30
32
Along length and width directions
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.3
0.6
0.9
1.2
1.5
40
41
41
42
42
43
43
44
44
Along length and width directions
Journal of Engineering Volume 13 September2006 Number 3
1717
Side view of the central section along length direction
Top view
Fig. (17): Final cracking pattern for slab cast in winter with B/L=0.4
Side view of the central section along length direction
Top view
Fig. (18): Final cracking pattern for slab cast in winter with B/L=0.8
Side view
of the central section along length direction
A.F. Ali Effect of Transverse Base Restraint on O. Al-Farouq Al-Damluji the Cracking Behavior of Massive Concrete
A. H. A. Al-Zuhairi
2021
Top view
Fig. (19): Final cracking pattern for slab cast in winter with B/L=1.0
REFRENCES
ACI Committee 207, “Effect of Restraint, Volume Change, and Reinforcement on Cracking in
Massive Concrete”, (ACI 207.2R-95). American Concrete Institute, Detroit, 1999, 26 pp.
ACI Committee 209, “Prediction of Creep, Shrinkage, and Temperature Effects in Concrete
Structures”, (ACI 209R-92), (Reapproved 1997). American Concrete Institute, Detroit, 1999, 47
pp.
ANSYS 5.4 Inc., “ANSYS Theory Reference”, Eighth Edition, SAS IP, Inc. 1997, Chapter 4 pp.
48-56.
Carlson, R. W., “Drying Shrinkage of Large Concrete Members” ACI Journal, Proceedings, Vol.
33, January-February 1937, pp.327-336.
Cook, R. D., Malkas, D. S., and Plesha, M. E., “Concepts and Applications of Finite Element
Analysis”, John Wiley and Sons Inc., Third edition, 1989, 630 pp.
Fung, Y. C., “Foundations of Solid Mechanics”, Prentice-Hall Inc., Englewood Cliffs, New Jersey,
1965, 525 pp.
Journal of Engineering Volume 13 September2006 Number 3
1719
Holman, J. P., “Heat Transfer”, Fourth Edition, 1976, McGraw-Hill.
Kammouna, Z. M., “Development of A Mathematical Model for Creep of Concrete with Reference
to Baghdad Climate”, M.Sc. Thesis, Baghdad University, College of Engineering, 2001, 90 pp.
Willam, K. J., and Warnke, E. D., “Constitutive Model for the Triaxial Behavior of Concrete”,
Proceedings, International Association for Bridge and Structural Engineering, Vol. 19, ISMES,
Bergamo, Italy, p. 174 (1975).
Journal of Engineering Volume 13 September2006 Number 3
0861
MONITORING PROCESS IN TURNING OPERATIONS FOR
CRACKED MATERIAL ALLOY USING STRAIN AND
VIBRATION SENSOR WITH NEURAL NETWORK
CLASSIFICATION
Asst.Prof.Dr.Nabeel Kadim Abid AL-Sahib
University of Baghdad Al-khawarizmi College of Engineering
Mechatronics Eng. Department
ABSTRACT
Surface finish and monitoring tool wear is essential for optimization of machining parameters
and performing automated manufacturing systems. There is a very close relationship between tool
wear and machining material parameters as surface roughness, shrinkage, cracks, hard particle ... etc.
Monitoring of manufacturing processes plays a very important role to avoid dawn time of the
machine, or prevent unwanted conditions such as chatter, excessive tool wear or breakage. Feature
extraction and decision making is a matter of considerable interest for condition monitoring of
complex phenomena with multiple sensors.
In this work, the implementation of a monitoring system utilizing simultaneous
vibration and strain measurements on the tool tip is investigated for the shrinkage and crack
of cast iron work piece. Machining parameters taken into consideration are cutting speed (116.5
and 136.6) m/min, feed rate (0.17 and 0.23)rev/min respectively and depth of cut (1) mm.
Data from the machining processes were recorded with one piezoelectric strain sensor type (PCB
740B02) and an accelerometer type (4370), each coupled to the data acquisition card type (9111 DR).
There were 22 features indicative of crack were extracted from the original signal. These include
features from the time domain (mean, STD, crest factor, RMS, kurtosis, variance), frequency
domains (power spectral density), time-series model coefficient (AR) and four packet features
extracted from wavelet packet analysis (RMS, STD, kurtosis, crest factor).
The (2x1) self organizing map neural network was employed to identify the crack and
shrinkage effect on the tool state. The program used with this process is MATLAB V.6.5. As a result
of the present work, we have an SOM model can classifying the crack with minimal error.
الخلاصة:
مراقبرة بري علاقرة كلره اان مثلر نظومرة الحير يع تشغيل السطوح ومراقبة بليان أداة القطع ضررويةة لحديةري الوررول ال
الر. بليان اداه القطع مع ظرول تشغيل سطوح المسبوكات المحشققة، الخشر ة، الم ممشرة والمدحوةرة علر اسريمات ورلية ومرا الر
برب دويتلعب عمليات الحي يعمراقبة الماك رة ، أو ةم رع طرروي يرروررل مهم اريا لح الحاكرل المررري مر وبرة مثرل الثرةررة ، وقر
لمراقبة الوواهر المعقية بالمدسسات المحعيدة ة انهمية إظحزاع وإتخاذ القرايات مسألة كبيروالمسر أللاداة
، تطبيق ظوام مراقبة ةسحعمل إهحزازا آظيرا ومقراةي إاهراد علر يألأ اداة مر وطرق قطعرة لإظممرا ادرر لحفي هذا العمل
( دوية ..11و 1100) تغذةة، ظسبة دقيقة/ م (0.818و 00811) القطع ب ور الإعحباي سرعة معاملات الحشغيل الديةي اليلب أخذت
ر عمليرات ا( مليمحرر 0عل الحروالي وعمرق القطرع ) دقيقة / ب روع مدسر إاهراد الحشرغيللبياظرات م ل piezoelectric (PCBسر
740B02( ل ميرزة مشطرر علر الشرق ..كران ه را .) DR 9111 (ظروع (، كل مزاوج إل بطاقة امع البياظرات4370( وظوع مع
( الإطاية اولية تحضم ميزات مال الوق (،mean, STD, crest factor, RMS, kurtosis, varianceإظحزع م
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
0860
رر تدليررل مزمررة ميبررع وا AR )) المحرروالي ال مرروذاي الررزم ثافررة ييررر كهربا،يررة(، معامررل مررانت تررردد )ك م يررزات مررزم إظحزعرر
wavelet (RMS, STD, kurtosis, crest factor ) 2) انx1 لحمييرز الإظممرا والشرق ( ةر وم طربمة عيربية إسرحخيم
، و Matlab V.6.5 بهذه العملية يمالمسحخ إن البرظامج عل مالة أداة القطع ةمم ر SOMظمروذج ميل ا علر ك حية للعمل الدالي
مم م خطأباقل أن ةي ر الشق
KEYWORDS :MONITORING, MEASUREMENT,WEAR,NEURAL NETWORK,CUTTING
,VIBRATION
SURVEY OF MONITORING PROCESS:
(Xiaoli etal 2000):used the wavelet transforms and fuzzy techniques are used to monitor tool
breakage and wear conditions in real time according to the measured spindle and feed motor currents ,
respectively .(Reuben etal 1998) : feature extraction and decision – making was a matter of consider
interest for condition monitoring of complex phenomena with multiple sensors. (Kndili etal 2003) :
have studied the outlines of a neural networks based modular tool condition monitoring system for
cutting tool wear classification . Multi layer neural network structure was used and data set has been
trained off – line using back propagation algorithm, an important variation in mean, RMS, standard
deviation of cutting forces and vibration can result in estimation and classification error. (Scheffer
etal 2001): discussed the implementation of a monitoring system utilizing simultaneous vibration and
strain measurement on the tool tip, was investigated for the wear manufacturing of aluminum pistons.
Data from the manufacturing process was recorded with two piezoelectric strain sensors and an
accelerometer, each coupled to a DSPT analyzer. A large number of features indicative of tool wear
automatically extracted from different parts of the original signals (Nadgir etal 2000): studied the out
line of a neural network based (TCMS) for cutting tool state classification. Orthogonal cutting tests
were performed on H13 steel using PCBN inserts and on line cutting force data was acquired with a
piezoelectric force dynamometer. Simultaneously flank wear data was measured using a tool makers
microscope and along with the processed data were fed a back propagation neural network to be
trained .All papers which discussed previously were used different method to measure the tool wear
such as using (vibration, strain, acoustic emission and feed current signal), the steel and aluminum
material was used in most papers as a work – piece with constant cutting condition. In this work are
using monitoring processes to classify the cutting tool wear through using the strain and vibration
signal , which measured from piezoelectric strain sensors and accelerometer respectively , after that
we extract the feature from the signal to applying it into SOM neural network to have the
classification of the amount tool wear . It can be divide in two parts theoretical included: (Time
domain, Modeling domain, Frequency domain, Wavelet packet coefficients) from the original signals,
and use it in SOM neural network. In addition classifying the cutting tool wear using SOM neural
network. And experimental such as: (Machining of Cast Iron shaft with and without crack on the
surface work-piece using turning machine with different cutting condition; Measuring the wear of
Carbide cutting tool type(SNMG 120412) using microscope; Measuring the signal from piezoelectric
strain sensor and accelerometer using Data Acquisition Card ).
WEAR IDENTIFICATION AND MONITORING
A conventional method for tool wear and shrinkage appeared at the work piece identification
is basically a two-step approach: First, extract features from the signals of a single sensor that
is highly sensitive to tool-wear and shrinkage but insensitive to noise; then, a physical model is
established to reflect the relationship between the sensor signals and the wear status. This is
because a crack model based on information from a single sensor cannot adequately reflect the
complexity of a cutting process. Therefore, approaches using sensor integration have been
Journal of Engineering Volume 13 September2006 Number 3
086.
introduced in the area of crack and shrinkage monitoring, and have attracted wide interest in recent
years. Cutting force measurement is one of the most commonly employed methods for on-line tool
and work piece state monitoring, especially in turning because cutting force values are more sensitive
to tool and work piece wear than other measurements such as vibration or acoustic emission. In
recent years, intelligent control systems such as, genetic algorithm, fuzzy logic, artificial neural
network and hybrid of these methods (fuzzy-neuron) are popular (Xiaoli 2000). Conventional
methods are mathematical model based systems, so it is difficult to take the machining
parameter into account in such models. But artificial intelligence based methods are less
dependent on the machine parameters.
MONITORING STAGES:
The classification of work piece and tool wear is a complex task because wear
introduces very small changes in a process with a very wide dynamic range. Furthermore, it is difficult
to identify whether a change in a signal is caused by wear or a change in the cutting
conditions. The task of wear monitoring can be subdivided into a number of stages (Rueben 1998):
Sensor selection and deployment.
Generation of a set of features indicative of wear condition.
Classification of the collected and processed information as to determine the amount of wear.
Sensors used in Monitoring Systems:
The sensors used for monitoring tool conditions can be divided into Two categories: direct and
indirect. Despite their high accuracy, direct sensors are rarely used in real-time industrial applications
because of their high cost and difficulty of installation also the direct measurements are not
possible in many instances such as drilling and milling. Further, such measurements in most
cases involve interruption of the machining process. On the other hand, indirect sensors, which
are relatively economical and small, can be used for on-line crack and shrinkage detection if a certain
relationship between sensor signals and tool-wear status can be established. A variety of indirect
sensing methods have been applied to crack monitoring studies, including cutting force signal
detection, cutting temperature detection, electrical resistance measurement, cutting vibration
detection, measurement of AE (Kandilli 2003) and electrical signals like spindle motor current and
power are also useful sources of information about tool states.
Wavelet Transform in Monitoring Process:
Signal processing is a very important step for (TCM). Recently, wavelet transform has
provided a significant new technique in signal processing, because it offers solution in the time-
frequency domain and is able to extract more information in the time domain at different frequency
band. There have been many research activities in the application WT for tool condition monitoring
(Scheffer 2001)
It uses wavelet transform to decompose measured signals. Acoustic emission signal and RMS
value of decomposed signals are taken as tool wear monitoring features.
It uses wavelet transform to analyze cutting force signals and wavelet transform
coefficients are taken as recognition parameters of crack.
Neural Networks in Monitoring Process:
In recent past, neural network models which employ cutting forces for estimation as well as
classification of wear have been developed. The present work outlines a neural networks based
modular tool condition monitoring system for crack and shrinkage classification. A multi layer neural
network structure was used and data set has been trained off-line using SOM algorithm. An important
variation in mean, RMS, standard deviation of cutting forces and vibration can result in estimation and
classification error.
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
086.
MONITORING PROCESS STAGES:
Work –Piece and Cutting Tool Materials:
Cemented carbide is composed of carbides tungsten, titanium and tantalum with some percentage
of cobalt. The chemical composition of work–piece material are given in Table ( 1 ) .
Table ( 1 ) chemical composition of cast Iron work piece .
C% Mn% Si% P% S% Cr% Ni% M% Fe
3.31 0.72 3.09 0.65 0.13 0.081 0.062 0.038 balance
Cutting Tool Wear Measurement:
After switching off the turning machine the cutting tool wear measured through the
microscope. The process of measuring (sensors signal and cutting tool wear) was repeated for other
W.P. that have the same dimensions until the wear level reach the maximum (0.3)mm , the cutting
condition of the machine were then changed , and the process was repeated . For each of the above
tests, strain and vibration data were obtained in order to be used in wear classification by neural
network techniques.
Carbide tools: roughing = 0.8 mm.
Carbide tools: finishing = 0.35 down to 0.15 mm.
Instrument Used in Monitoring Process:
Different instruments are used for different monitoring process depending on the variable to be
measured. In the present work (a piezoelectric strain sensor Model 740B02 used to measure the strain
at the tool holder and this gives an indication of cutting force applied to the cutting tool , signal
conditioner for strain sensor is used for amplifying and analyzing signals , Accelerometer sensor is
used to measure the vibration at the tool holder , The power amplifier is used to amplify the
accelerometer signal , Data Acquisition Card type PCI-9111 , used for signal analysis application and
process control , Microscope used in the measurement of tool wear , Interface between the instrument
and the turning machine) .
The system ready to measure the data , from the turning machine as shown in Fig ( 1 ) .
Experimental Design of Monitoring Processes:
A set of tool wear cutting data were acquired by machining a bar of Cast Iron under a given
set of cutting conditions with a coated cemented carbide tip Table (2). The set of sensors
used, were an accelerometer for measuring vertical vibrations, piezoelectric strain sensor on tool
holder for force measurement as a strain, in order to find the amount of crack compared with the
amount of strain and vibration signal. The specification of the instruments used, listed in Table (3)
Journal of Engineering Volume 13 September2006 Number 3
0861
Table (2): Experimental parameters used in the monitoring process
Table (3) Instruments used in the monitoring process
The turning operation was carried out using (HARRISON 15”) turning machine. The
experimental set-up and instrumentation are shown in Fig (1). Most previous work interested in
monitoring process for crack found the sampling rate at 10 KS/s enough to represent the
analogue signal for cutting force. In this work the analogue signals were sampled with an
Ampilicon PCI 9111DR data- acquisition board at a sample rate of 10K Hz per channel for a time
period of 25.6 ms, number of samples reading through each stage are 12800 sampling pear
channel. Data were acquired at intervals between (1.5~3.5 ) min depending on cutting
conditions at which point crack was also measured, taking into account tool life inserts. The total
number of tests is 2 each having different cutting conditions (to construct test and conformation
sets). Each data record, of 12800 points acquired at the end of the cut, was processed to
generate features used in the classification stage. Each feature vectors were extracted from time
domain, wavelet domain, frequency domain, and model domain of all sensors. These features were
then passed directly to the neural networks for classification (Silva 2000), with the training data
coming from selected tests and the testing data used being from tests that were not used during the
training phase. The cutting conditions used during the training experiment see in Table ( 4 ).
Table (4) Cutting conditions used in machining experiment .
Description Components HARRISON 15" Lathe Cast iron shaft (D:70mm L:270mm) Work piece Sandvik SDJCR 2020K11 Holder type Carbide tips sintered square
SNMG 12041 pattern TP15
120412 ) pattern grade (TP15)
Insert type
0.078 -0.23 mm/rev Feed rate 105-165m/min Cutting speed 1 mm Depth of cut
Tool life for
each test (min)
Number of
components Wear land (mm) Depth of cut
DOC (mm)
Feed mm/rev
Cutting speed m/min
Diameter of shaft (mm)
Test No
17 13 0.033-0.21 1 0.23 136.6 58 1
13 6 0.002-0.44 1 0.17 116.5 70 2
Mounting Description Sensor On tool holder PCB model
740B02 Piezoelectric strain sensor
On tool holder Type 4370 Accelerometer Type PCB
480E09
Signal conditioner
Type 2626 Power amplifier In computer board PCI 9111DG Data acquisition card
Math lab V6.5 Software program
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
0861
Test (1 and 2) with constant DOC .
Fig (1) Experimental set-up and Instrumentation
1. Cast iron shaft with dimension ( D:70 mm, L:270mm)
2. Square carbide tool
3. Accelerometer
4. Strain sensor
5. Tool holder
6. Power amplifier type 2626
7. Signal conditioner type PCB 480E02
8. DAQ card type 9111DR installed in PC board
9. PC P4 installed MATLAB program
Programming of Monitoring Process :
The flow chart of the programming process is shown in Fig.(2).
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0868
Fig ( 2 ) flow chart of the programming process .
Feature Extraction
To increase the reliability of the tool wear monitoring system, in the presented work a
monitoring strategy was devised that is based on four type of feature:
Time and modeling domain feature
Frequency domain feature
Wavelet packet analysis feature
The Features of the Time and Modeling Domain:
We can use some time – domain features as descriptors of crack and shrinkage (Nadgir 2000),
therefore, the following time-domain features were extracted from each signal: mean, rms,
crest factor, standard deviation, skewness and kurtosis. A brief mathematical description of each is
given as follows:
1- mean: the mean value of a function x(t) over an interval N is
Original
signal
Features
Time domain
features
Model coefficient
features
Frequency
domain feature
Choose four packets
containing most energy
-Rms
-Mean
-Kuntosis
-Crest factor
-Variance
-Skweness
Power spectral
density for 8.5
KHz
-Auto regressive
-Moving average
-Auto regressive
/moving average
On chosen packet
- rms
- kuntosis
- crest factor
- standard deviation
Choose for best features based
on high correlation
Wear Classification som
Wavelet packet
analysis
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
0860
(1)
2- standard deviation (б) :
(2)
3- root mean square (rms) : the rms value Xrms of a function x(t) over an interval of N is :
(3)
4- the crest factor CF: is the ratio of the peak level(Xmax ) to the RMS level (Xrms )
(4)
5- the skewness S : is the third statistical moment of a distribution :
(5)
6- the kurtosis K : is the fourth statistical moment of distribution :
(6)
Time-series models of the sensor signals are constructed and the model coefficients are
used as features indicative of crack (Kuo 1999, Ravindra 1997 and Obikawa 1996). This is
because the model coefficients represent the characteristic behavior of the signal. Depending on the
order of the model, a number of model coefficients can be chosen. Normally, only the first
model coefficient, or sometimes the first three to four model coefficients are chosen, because
they are most descriptive of the signal characteristics of interest. In this case coefficients from
(AR) models were considered. A brief discussion of these models follows (Rueben 1998).
In a pth-order AR model for a time series x (n), where n is the discrete time index, the
current value of the measurement is expressed
as a linear combination of p previous values:
x(n)= a1 x(n-1)+ a2 x(n-2)+……….ap x(n-p) (7)
Where a1, a2, a3… ap are the AR coefficients. The first AR coefficient was chosen as a feature.
Frequency Domain Feature:
The most common frequency domain characteristic in the literature is the spectral energy
around the first natural frequency of the tool-work piece system (Rueben 1998). It was established
Journal of Engineering Volume 13 September2006 Number 3
0866
that the fundamental natural frequency for this system lies at about 8.5 kHz. The spectral energy at
8.5 KHz was taken as a feature.
(8)
With sx(f) the one-sided PSD function and fl and fh the lower and upper frequencies chosen to
reflect the energy in the region of interest.
Wavelet Packet Analysis Feature
The wavelet transform is a relatively new method of signal processing that has been applied to
many engineering studies with great success. Fairly recent studies also proved that wavelet analysis
could be utilized for monitoring of the machining process (Jiang 1987). The success of the wavelet
transform is generally attributed to the natural shape of the wavelet, which is more descriptive of
most natural processes than the sine function used in Fourier analysis. Wavelet analysis is capable of
revealing aspects of data that other signal analysis techniques miss, like trends, breakdown points,
discontinuities in higher derivatives, and self-similarity. In this instance, wavelet packet analysis was
used to generate features that may show consistent trends towards tool and work piece wear. Like
other wavelet analysis techniques such as (DWT), wavelet packet analysis also requires the
construction of a wavelet decomposition tree. Each packet in the decomposition tree contains
information on the original signal in the form of wavelet coefficients. The original signal can be
reconstructed using any chosen number of the packets on the tree. However, the normal practice is to
choose the packets containing the most information on the original signal, and then discarding the
packets containing noise or less important information. Usually, an energy-based approach is used to
choose the optimal packets. The Shannon entropy formula was used, see equation (9), which is a
non-normalized entropy involving the logarithm of the squared value of each signal sample or, more
formally (Obikawa 1996),
(9)
Where E is the Shannon entropy and Si is the signal sample at instant i.
In this study, the method requires that a reliable wavelet packet analysis be established for the
given signal. The reliability of the wavelet packet analysis can be investigated in a number of ways,
such as assessing the cross-correlation, rms error and cross-coherence between the original signal and
the reconstructed signal. A number of packets containing the most energy representative of the
original signal must then be chosen. The order of the decomposition tree will determine the
maximum number of representative packets that may be chosen.
Wear classification using neural network (SOM):
The self-organizing maps, developed by Kohonen (Kohonen 1998), is a fairly new and
effective software crack for data analysis. The SOM has been implemented successfully in
numerous applications, in fields such as process analysis, machine perception, control and
communication (Surender 1994). The SOM implements the orderly mapping of high-dimensional
data onto a regular low-dimensional grid. Thereby the SOM is able to identify hidden relationships
between high-dimensional data into simple geometric relationships that can be displayed on a
simple figure. The SOM can generally be described as a neutral network with self-organizing
capabilities. Most neural networks require information and interaction from the user for
classification. Although the SOM was intended as a data visualization tool, it can be
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
0861
used for classification as well. The SOM automatically arranges the data on a two-dimensional grid
of neurons where similar observations are placed close to one another and dissimilar ones further
away. If the classes of some of the observations are known, certain regions on the grid could be
allocated for these classes. The computation of the SOM is a non-parametric, recursive
regression process. The incremental-learning SOM algorithm can be described briefly as
mi(t + 1) = mi(t) + hc(x),i(x(t) – mi(t)) (10)
Where t is the index of the regression step, and the regression is performed recursively
for each presentation of a sample of x, denoted x (t). The scalar multiplier hc(x),i is called
the neighborhoods function, which causes similar observations to be placed in the same region on
the map. Its first subscript c = c(x) is defined by the condition
(11)
which means that mc (t) is the model that matches best with x(t). The neighborhoods function is
often taken as the Gaussian function. Note that a batch version of algorithm exist which is
computationally much faster.
RESULT AND DISCUSSIONS
Flank Wear Accursed in Machine Cutting Tool due to Crack
The sudden flank wear for tests (1) are shown in Fig (3). Certain features of flank wear are
identified, first an extreme condition of flank wear often appears on the cutting edge at locations
corresponding to the original surface of the work piece it is called (notch wear) .It is accrue because
the original work surface is harder and more abrasive than the internal material due to sand particles
in the surface from casting or other reason. As a consequence of the harder surface, wear is
accelerated at this location. At this level of the tool wears, when the machining process continue the
fracture was increased, very high noise appeared and surface roughness of the work piece became
very bad (Surender 1994).
Fig (3) sudden wear at cutting tool due to presenting crack in work piece ,
wear land (0.33 - 0.2)mm (test1).
Journal of Engineering Volume 13 September2006 Number 3
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Effect of Cutting Condition Tool Life :-
Table (4) represent test with constant D.O.C and different cutting conditions , gave loot life of
(13 , 17) min respectively . from these tests it could be seen the tool life decrease with decrease in
cutting speed and feed rate . It is occur because the original work surface crack and harder particles
in the surface . As a consequence of crack or hard particles , wear is accelerated at this location .
Properties of Signals in Tool Condition Monitoring:
In the metal cutting, the signal from (strain and vibration) sensor is a transient energy
spontaneously released in material undergoing deformation or fracture or both. At microscopic level,
signal is related to grain size ,dislocation density ,and distribution of the second-phase particles
crystal-line form (Kannatey 1982) .The signals from (strain and vibration) sensor generator during
cutting operation are non stationary and may pass a different magnitude , damping ,frequency and
phase (Stern 1971 and Du 1991).
Fig (4) shows the sample of signals for (strain and vibration) sensor. The signal generated from
the data sample at 25600 Hz at 1 second contains 25600 data point, for both strain and vibration
signal from a turning operation. The continuation of the signal is contributed for by deformation of
work piece material at shear zones, the friction at tool work piece and chip –tool contact regions. In
Fig (5) the approximately constant amplitude signal that run throughout the record, constitute the
continuous part of the signals. Superimposed on the continuous part, the transient part of the signal is
generated by micro-cracks of the crystal structure of work piece material, nonhomogeneity of work
piece material and chip breakage. The high amplitude short –duration signals that appears in all
figures are the transient parts of the signal (Kamarthi 1997).
The signal generated from a machining process fundamentally depends on properties of tool
and work piece materials applied stress, strain rate and material volume involved in the deformation
process. In cutting process, the signal form the function between the tool and
work piece can be distinguished from the signals generated by the chip-breakage and the
material deformation process at the share zones this can be shown in Fig (5), where the chip-
breakage generated no uniform signal while the material deformation zone make the signal be
approximately stable and this can be shown clearly in the feature curves because the chip breakage
make the curve flow up or down instantaneously.
Fig (4) Samples of vibration and strain signals
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
0810
Fig (5) vibration signals for 300 sampling no.
Strain Signals Measured During the Tests and It’s Effect of Crack
As the signal spectral is sensitive to tool wear and tool fracture, it is possible to use signal for
tool condition monitoring, the strain signal as shown in Fig (4) is basically sinusoidal in nature.
During the course of their propagation, they often undergo considerable changes to scattering by
structural defects, multiple reflect at interface and refraction where there is a medium change along
the travel path (Kamarthi 1997). Fig (6) which show the strain signal measured for cutting tool in
test 1 with cutting speed 136.6 mm/sec and feed 0.23 mm/rev when the crack appear on the work
piece. Fig(6a) shown clearly that the strain signal have an greatest effect when the crack accurse ,
where the signal have an negative shoot at the lower frequency of the signal between ( 1000-3000)
Hz. Fig (6b) shows the same signal when the is no crack accursed in work piece where there is no
negative shoot in the strain signal. Fig (7) shows the power spectral density for the same which
shown the signal have a high constricted at the lower frequency when the crack at the work-piece
accursed. This produce an indication that when the crack appeared in the work piece the strain signal
have an negative shoot indication at the lower frequency of the signal
(a) (b)
Fig(6) a: strain signal with crack
b: strain signal without crack.
Journal of Engineering Volume 13 September2006 Number 3
081.
(a) (b)
Fig (7) a: PSD for strain signal with crack
b: PSD for strain signal without crack.
(a) (b)
Fig (8) a: vibration signal with crack
b: vibration signal without crack.
Vibration Signals Measured During the Tests and Its Effect of Crack
The vibration signal has a high effect on strain signal during the machining process. As can be
show in Fig (8a) where the crack appear at the work piece and compare the result with Fig (8b)
where the crack are not appear we can found there is no trend of vibration signal at the crack of the
work piece, this is because the vibration signal result from many external recourses such as gear of
cutting machine and other dynamic influence which make the lower frequency not sensitive at the
crack in work piece. Also Fig (9) which shows the PSD for the vibration signal in both state not give
any pure indication towered the crack.
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
081.
(a)
(b)
Fig (9) a: PSD for vibration signal with crack
b: PSD for vibration signal without crack.
Crack Effects at the Feature Extracted Form Measured Signal
All the feature which are extracted from the measured signals and used to recognized the effect
of crack at the strain and vibration signal can be shown in Table (5). When analyzing these feature it
can be found that the strain feature has a greatest trend toward crack than the vibration signal. Other
that there is a feature has good trend towered crack that others. The time features for strain signal
such as ( mean, STD, RMS) given a good indication toward crack where it’s values when the crack
found lower than when the crack disappear. Wavelet packet analysis feature have good indication for
crack, such as wavelet packet for (STD, S, K, RMS). In wavelet packet we know it’s divide the
signal in lower and highest frequency and then select the lower frequency which give good indication
for the signal behavior, so in this work we choose the wavelet packet (1 and 3) to represent the effect
of crack in wavelet packet domain, in modeling domain feature the auto regressive have good
indication towered crack where it’s value where the crack happened lower than when the crack
disappear. All features for strain signal have the same behavior for crack which it’s value with
Journal of Engineering Volume 13 September2006 Number 3
0811
presented the shrinkage and crack are lower than it’s value when the crack disappear. For the
vibration signal all the feature have lower trend toward crack where it’s values with present crack
approximately the same without present the crack.
Table (5) features values for strain and vibration signal measured during machining
process
Vibration signal
without crack
Vibration signal
with crack
Strain signal
without crack
Strain signal with
crack
features
2.8035 2.5952 51.477 1.3612 Mean 351.06 358.33 148.1 9.735 STD 217.16 201.02 3987.4 105.44 RMS 5.801 6.9225 0.16752 0.35845 CF
1.1244e+008 2.2063e+007 8.4593e+013 1.0934e+013 S 9.8614e+011 1.1244e+011 6.7478e+019 4.41e+018 K 170.09 175.79 109.39 10.146 WS1 466.55 475.17 178.53 9.2921 WS2 129.63 127.55 123.92 13.143 WS3 202.49 213.85 92.217 5.6827 WS4 154.36 143.29 2825.5 75.119 WR1 447.43 532.37 87.388 2.242 WR2 109.8 103.16 2007 53.92 WR3 109.99 46.03 6.0201 2.7417 WR4 5.3746 4.744 0.15768 0.31056 W1CF 3.5868 3.238 8.0926 14.862 W2CF 4.2721 5.193 0.22931 0.4681 W3CF 6.3551 14.407 55.881 15.186 W4CF 1.3137e+064 1.188e+062 9.964e+063 2.0635e+062 WK1 9.5615e+062 4.1267e+062 3.6012e+065 9.9748e+069 WK2
1.866e+062 7.2489e+067 1.4139e+064 4.1267e+062 WK3 5.4921e+062 4.6772e+058 1.0638e+060 1.0767e+064 WK4 0.80926 0.79146 0.37441 -0.089601 A(2)
Self Organizing Map for Neural Network:
The selected features from the two learning data sets were used to train (2 x 1) SOM with
10000 epoches. Figs (10-12) show the feature of SOM neural network for test 1. The SOM layer for
specific feature represent the behavior of that feature curve, there is a two specific layer for crack
state. The reason why only a small number of neurons are used is because it makes classification
easier (although less flexible). In this case, one neuron is used to correspond to no crack present and
one neuron refers to crack presented see Fig (13). When more neurons are used in the network, the
regions corresponding to a certain classification become larger, and classification becomes more
flexible, because when we increase the number of layer in the output the range of each feature to
have a specific layer will decrease so the layer have an precise specific feature. For each of the
selected features, a (2 x 1) representative SOM for test 1 can be shown in Figs (10 - 12). It is
important to note that although a SOM for each feature is available, the SOM is actually a single
entity. A view on a selected feature is only the view in the direction of that dimension. The SOM can
represent multidimensional data in this manner. This is illustrated in Figs (10-12), where all the
selected variables are shown on a single graph. When color coded, such a figure can display how the
values of the features correspond among one another. The observations in the (testing) data set were
labeled (no crack) and (crack), corresponding to the number of machined components (work piece
used in machining process). The best matching units for these data were looked up on the SOM. As
shown in Fig (13), it is clear that neurons 1 correspond to a no crack present and neuron 2 to a rack
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
0811
present. The figures show a SOM layer which distributed corresponding to the best matching units of
the test data. It is clear that the trajectory moves in time from the (without crack) to the (crack).
Fig (10) SOM for time domain feature for strain signal test (1).
SOM for STD for SOM for STD for
Wavelet packet (1) Wavelet packet (3)
Journal of Engineering Volume 13 September2006 Number 3
0818
SOM for RMS for SOM for RMS for
Wavelet packet (1) Wavelet packet (3)
SOM for CF for SOM for CF for
Wavelet packet (1) Wavelet packet (3)
Fig (11) SOM for wavelet packet analysis features for strain signal test (1).
Fig (12) SOM for Auto regressive at strain signal for test (1).
Fig (13) SOM for Crest Factor at strain signal for test (1) with label.
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
0810
CONCLUSIONS
1- The monitoring system can extract and select features quickly enough to enable the
manufacturer to implement an on-line monitoring system.
2- The result we showed that vibration signal has a lower trend toward crack than a strain signal
during a machining process.
3- Most features give approximately a behavior toward crack, that when the crack present the
features vales became lower values than when the crack not present.
4- RMS, STD, mean, PSD for 8.5 KHz, wavelet features for packet (3), it gives a better
indication of tool wear than other features.
5- When using SOM neural network a best correct classification of the tool can be obtained.
6- There are a cutting condition ( feed rate & cutting speed )with constant depth of cut in
machining process , which give good indication result for tool life and number of work – piece
( with and without crack) used.
7- A vibration signal has a high effect on strain signal during a machining process.
REFERENCES
Du R. and Yan 1991 .”time- frequency distripution of Acoustic Emission Signal for tool wear
detection in turning”.
C. Y. Jiang, Y.Z. Zhang and H. J. Xu 1987 ,” In-Process monitoring of tool wear stage by the
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S.V.Kamarthi.1997 “Flank wear estimation in turning through wavelet representation of acoustic
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I. Kandilli and H.Metin 2003. “development of On-Line monitoring system for tool wear in cutting
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T. Kohonen 1998 Neuro computing 21, 1-6. The self-organising map.
R. J. Kuo and P. H. Cohen 1999 “Multi-sensor integration for online tool wears estimation through
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A. Nadgir and Tugrul Özel. July 30-August 2, 2000, “Neural network modeling of flank wear for tool
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Notification:
A cross section area of shaft.(mm2) N Number of sampling
Ac alternative current.(A) n1 Net input in neural network
A/D analogue to digital converter Pc personal computer
AE acoustic emission PSD power spectral density
AR auto regressive R input vector
ARMA auto regressive moving average
ANN adaptive neural network RMS Root mean square
Rn end condition parameter S Skew ness function
BPNN Back propagation neural network
C SOM function SOM Self organizing map
Cf crest factor SOMF Self organizing map function
DAQ Data Acquisition Card T Time (sec.)
DSPT Digital signal processing transformer
DWT Discrete wavelet transform TCM Tool condition monitoring
F feed rate (mm/min)
f frequency function VB Vibration signal
(n) wavelet function X Mean value
h(n) wavelet function
ICP Integrate circuit programming
N.K. Abid AL-Sahib Monitoring Process in Turning Operations for Cracked
Material Alloy Using Strain and Vibration Sensor with
Neural NETWORK Classification .
0811
I.D.D Independent Identification Distribution
K kurtosis function
L length shaft
MA Moving average